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Theorem clnbgrgrim 49247
Description: Graph isomorphisms between hypergraphs map closed neighborhoods onto closed neighborhoods. (Contributed by AV, 2-Jun-2025.)
Hypothesis
Ref Expression
clnbgrgrim.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
clnbgrgrim ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝐻 ClNeighbVtx (𝐹‘𝑋)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑋)))
Proof of Theorem clnbgrgrim
Dummy variables 𝑒 𝑔 𝑖 𝑗 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6920 . . . . . . . . . . . 12 (𝑛 = 𝑋 → ((𝐹‘𝑛) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
2 clnbgrgrim.v . . . . . . . . . . . . . 14 𝑉 = (Vtx‘𝐺)
32clnbgrvtxel 49142 . . . . . . . . . . . . 13 (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
433ad2ant3 1153 . . . . . . . . . . . 12 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
5 eqidd 2764 . . . . . . . . . . . 12 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) = (𝐹‘𝑋))
61, 4, 5rspcedvdw 3583 . . . . . . . . . . 11 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = (𝐹‘𝑋))
76adantr 486 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = (𝐹‘𝑋))
8 eqeq2 2775 . . . . . . . . . . 11 (𝑥 = (𝐹‘𝑋) → ((𝐹‘𝑛) = 𝑥 ↔ (𝐹‘𝑛) = (𝐹‘𝑋)))
98rexbidv 3190 . . . . . . . . . 10 (𝑥 = (𝐹‘𝑋) → (∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥 ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = (𝐹‘𝑋)))
107, 9syl5ibrcom 250 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → (𝑥 = (𝐹‘𝑋) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
11 simpl2 1211 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
12 simpl1 1210 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
13 simp3 1156 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
14 simpl 488 . . . . . . . . . . . . 13 ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) → 𝑥 ∈ (Vtx‘𝐻))
1513, 14anim12i 625 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → (𝑋 ∈ 𝑉 ∧ 𝑥 ∈ (Vtx‘𝐻)))
16 eqid 2763 . . . . . . . . . . . . 13 (Vtx‘𝐻) = (Vtx‘𝐻)
17 eqid 2763 . . . . . . . . . . . . 13 (Edg‘𝐻) = (Edg‘𝐻)
182, 16, 17clnbgrgrimlem 49246 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝑋 ∈ 𝑉 ∧ 𝑥 ∈ (Vtx‘𝐻))) → ((𝑒 ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑋), 𝑥} ⊆ 𝑒) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
1911, 12, 15, 18syl3anc 1399 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → ((𝑒 ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑋), 𝑥} ⊆ 𝑒) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
2019expd 421 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → (𝑒 ∈ (Edg‘𝐻) → ({(𝐹‘𝑋), 𝑥} ⊆ 𝑒 → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥)))
2120rexlimdv 3165 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → (∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒 → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
2210, 21jaod 873 . . . . . . . 8 (((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ (𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻))) → ((𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
2322expimpd 459 . . . . . . 7 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
24 eqid 2763 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
25 eqid 2763 . . . . . . . . 9 (iEdg‘𝐻) = (iEdg‘𝐻)
262, 16, 24, 25grimprop 49196 . . . . . . . 8 (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
27 f1of 6849 . . . . . . . . . . . . . . . 16 (𝐹:𝑉⟶1-1onto(Vtx‘𝐻) → 𝐹:𝑉⟶(Vtx‘𝐻))
2827adantr 486 . . . . . . . . . . . . . . 15 ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → 𝐹:𝑉⟶(Vtx‘𝐻))
29283ad2ant1 1151 . . . . . . . . . . . . . 14 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → 𝐹:𝑉⟶(Vtx‘𝐻))
3029ad2antrr 739 . . . . . . . . . . . . 13 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → 𝐹:𝑉⟶(Vtx‘𝐻))
312clnbgrisvtx 49143 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑛 ∈ 𝑉)
3231adantl 487 . . . . . . . . . . . . . 14 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) → 𝑛 ∈ 𝑉)
3332adantr 486 . . . . . . . . . . . . 13 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → 𝑛 ∈ 𝑉)
3430, 33ffvelcdmd 7112 . . . . . . . . . . . 12 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → (𝐹‘𝑛) ∈ (Vtx‘𝐻))
35 eleq1 2852 . . . . . . . . . . . . . 14 (𝑥 = (𝐹‘𝑛) → (𝑥 ∈ (Vtx‘𝐻) ↔ (𝐹‘𝑛) ∈ (Vtx‘𝐻)))
3635eqcoms 2771 . . . . . . . . . . . . 13 ((𝐹‘𝑛) = 𝑥 → (𝑥 ∈ (Vtx‘𝐻) ↔ (𝐹‘𝑛) ∈ (Vtx‘𝐻)))
3736adantl 487 . . . . . . . . . . . 12 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → (𝑥 ∈ (Vtx‘𝐻) ↔ (𝐹‘𝑛) ∈ (Vtx‘𝐻)))
3834, 37mpbird 260 . . . . . . . . . . 11 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → 𝑥 ∈ (Vtx‘𝐻))
39 simp3 1156 . . . . . . . . . . . . 13 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
4029, 39ffvelcdmd 7112 . . . . . . . . . . . 12 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ (Vtx‘𝐻))
4140ad2antrr 739 . . . . . . . . . . 11 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → (𝐹‘𝑋) ∈ (Vtx‘𝐻))
42 eqid 2763 . . . . . . . . . . . . . . . 16 (Edg‘𝐺) = (Edg‘𝐺)
432, 42clnbgrel 49141 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑛 = 𝑋 ∨ ∃𝑘 ∈ (Edg‘𝐺){𝑋, 𝑛} ⊆ 𝑘)))
44 fveq2 6911 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑋 → (𝐹‘𝑛) = (𝐹‘𝑋))
4544orcd 887 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑋 → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))
46452a1d 27 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑋 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
4724uhgredgiedgb 29737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐺 ∈ UHGraph → (𝑘 ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)𝑘 = ((iEdg‘𝐺)‘𝑗)))
4847adantr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑘 ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)𝑘 = ((iEdg‘𝐺)‘𝑗)))
49483ad2ant2 1152 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (𝑘 ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)𝑘 = ((iEdg‘𝐺)‘𝑗)))
5049biimpa 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) → ∃𝑗 ∈ dom (iEdg‘𝐺)𝑘 = ((iEdg‘𝐺)‘𝑗))
51 2fveq3 6916 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑖 = 𝑗 → ((iEdg‘𝐻)‘(𝑔‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑗)))
52 fveq2 6911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑖 = 𝑗 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑗))
5352imaeq2d 6065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑖 = 𝑗 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗)))
5451, 53eqeq12d 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑖 = 𝑗 → (((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))))
5554rspcv 3576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑗 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))))
56553ad2ant3 1153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))))
57 sseq2 3939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 ↔ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗)))
58573ad2ant3 1153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗)) ∧ 𝑘 = ((iEdg‘𝐺)‘𝑗)) → ({𝑋, 𝑛} ⊆ 𝑘 ↔ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗)))
59 sseq2 3939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑒 = ((iEdg‘𝐻)‘(𝑔‘𝑗)) → ({(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒 ↔ {(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ ((iEdg‘𝐻)‘(𝑔‘𝑗))))
6025uhgrfun 29677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
6160adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → Fun (iEdg‘𝐻))
62613ad2ant3 1153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) → Fun (iEdg‘𝐻))
63 f1of 6849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) → 𝑔:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
6463adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → 𝑔:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
65643ad2ant1 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) → 𝑔:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
6665ffvelcdmda 7111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐻))
6725iedgedg 29661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((Fun (iEdg‘𝐻) ∧ (𝑔‘𝑗) ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) ∈ (Edg‘𝐻))
6862, 66, 67syl2an2r 698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) ∈ (Edg‘𝐻))
69683adant2 1149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) ∈ (Edg‘𝐻))
7069adantr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) ∈ (Edg‘𝐻))
71703ad2ant1 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) ∈ (Edg‘𝐻))
72 f1ofn 6850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (𝐹:𝑉⟶1-1onto(Vtx‘𝐻) → 𝐹 Fn 𝑉)
7372adantr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → 𝐹 Fn 𝑉)
74733ad2ant1 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) → 𝐹 Fn 𝑉)
75743ad2ant1 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → 𝐹 Fn 𝑉)
7675adantr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) → 𝐹 Fn 𝑉)
77 pm3.22 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
7876, 77anim12i 625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
79783adant2 1149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
80 3anass 1111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝐹 Fn 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
8179, 80sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
82 fnimapr 6994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝐹 Fn 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝐹 “ {𝑋, 𝑛}) = {(𝐹‘𝑋), (𝐹‘𝑛)})
8381, 82syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝐹 “ {𝑋, 𝑛}) = {(𝐹‘𝑋), (𝐹‘𝑛)})
84 imass2 6068 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ({𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) → (𝐹 “ {𝑋, 𝑛}) ⊆ (𝐹 “ ((iEdg‘𝐺)‘𝑗)))
85843ad2ant2 1152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝐹 “ {𝑋, 𝑛}) ⊆ (𝐹 “ ((iEdg‘𝐺)‘𝑗)))
8683, 85eqsstrrd 3948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → {(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ (𝐹 “ ((iEdg‘𝐺)‘𝑗)))
87 simp1r 1217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗)))
8886, 87sseqtrrd 3950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → {(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ ((iEdg‘𝐻)‘(𝑔‘𝑗)))
8959, 71, 88rspcedvdw 3583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) ∧ {𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)
90893exp 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗))) → ({𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
91903adant3 1150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗)) ∧ 𝑘 = ((iEdg‘𝐺)‘𝑗)) → ({𝑋, 𝑛} ⊆ ((iEdg‘𝐺)‘𝑗) → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
9258, 91sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗)) ∧ 𝑘 = ((iEdg‘𝐺)‘𝑗)) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
93923exp 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → (((iEdg‘𝐻)‘(𝑔‘𝑗)) = (𝐹 “ ((iEdg‘𝐺)‘𝑗)) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))
9456, 93syld 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) ∧ 𝑋 ∈ 𝑉 ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))
95943exp 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) → (𝑋 ∈ 𝑉 → (𝑗 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))))
9695com34 92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) ∧ 𝑘 ∈ (Edg‘𝐺) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph)) → (𝑋 ∈ 𝑉 → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))))
97963exp 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → (𝑘 ∈ (Edg‘𝐺) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑋 ∈ 𝑉 → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))))))
9897com25 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ 𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑋 ∈ 𝑉 → (𝑘 ∈ (Edg‘𝐺) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))))))
9998expimpd 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐹:𝑉⟶1-1onto(Vtx‘𝐻) → ((𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑋 ∈ 𝑉 → (𝑘 ∈ (Edg‘𝐺) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))))))
10099exlimdv 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹:𝑉⟶1-1onto(Vtx‘𝐻) → (∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑋 ∈ 𝑉 → (𝑘 ∈ (Edg‘𝐺) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))))))))
101100imp 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑋 ∈ 𝑉 → (𝑘 ∈ (Edg‘𝐺) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))))))
1021013imp 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (𝑘 ∈ (Edg‘𝐺) → (𝑗 ∈ dom (iEdg‘𝐺) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))))
103102imp31 423 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) ∧ 𝑗 ∈ dom (iEdg‘𝐺)) → (𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
104103rexlimdva 3167 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) → (∃𝑗 ∈ dom (iEdg‘𝐺)𝑘 = ((iEdg‘𝐺)‘𝑗) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
10550, 104mpd 16 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
106105ex 418 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (𝑘 ∈ (Edg‘𝐺) → ({𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
107106com14 97 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑘 ∈ (Edg‘𝐺) → ({𝑋, 𝑛} ⊆ 𝑘 → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
108107imp 412 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) → ({𝑋, 𝑛} ⊆ 𝑘 → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
1091083imp 1128 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) ∧ {𝑋, 𝑛} ⊆ 𝑘 ∧ ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉)) → ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)
110109olcd 888 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) ∧ {𝑋, 𝑛} ⊆ 𝑘 ∧ ((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉)) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))
1111103exp 1137 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑘 ∈ (Edg‘𝐺)) → ({𝑋, 𝑛} ⊆ 𝑘 → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
112111rexlimdva 3167 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (∃𝑘 ∈ (Edg‘𝐺){𝑋, 𝑛} ⊆ 𝑘 → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
113112com12 33 . . . . . . . . . . . . . . . . 17 (∃𝑘 ∈ (Edg‘𝐺){𝑋, 𝑛} ⊆ 𝑘 → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
11446, 113jaoi 871 . . . . . . . . . . . . . . . 16 ((𝑛 = 𝑋 ∨ ∃𝑘 ∈ (Edg‘𝐺){𝑋, 𝑛} ⊆ 𝑘) → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))))
115114impcom 413 . . . . . . . . . . . . . . 15 (((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑛 = 𝑋 ∨ ∃𝑘 ∈ (Edg‘𝐺){𝑋, 𝑛} ⊆ 𝑘)) → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
11643, 115sylbi 220 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝐺 ClNeighbVtx 𝑋) → (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
117116impcom 413 . . . . . . . . . . . . 13 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))
118117adantr 486 . . . . . . . . . . . 12 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))
119 eqeq1 2767 . . . . . . . . . . . . . . 15 (𝑥 = (𝐹‘𝑛) → (𝑥 = (𝐹‘𝑋) ↔ (𝐹‘𝑛) = (𝐹‘𝑋)))
120 preq2 4695 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝐹‘𝑛) → {(𝐹‘𝑋), 𝑥} = {(𝐹‘𝑋), (𝐹‘𝑛)})
121120sseq1d 3944 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐹‘𝑛) → ({(𝐹‘𝑋), 𝑥} ⊆ 𝑒 ↔ {(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))
122121rexbidv 3190 . . . . . . . . . . . . . . 15 (𝑥 = (𝐹‘𝑛) → (∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒))
123119, 122orbi12d 932 . . . . . . . . . . . . . 14 (𝑥 = (𝐹‘𝑛) → ((𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒) ↔ ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
124123eqcoms 2771 . . . . . . . . . . . . 13 ((𝐹‘𝑛) = 𝑥 → ((𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒) ↔ ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
125124adantl 487 . . . . . . . . . . . 12 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → ((𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒) ↔ ((𝐹‘𝑛) = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), (𝐹‘𝑛)} ⊆ 𝑒)))
126118, 125mpbird 260 . . . . . . . . . . 11 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒))
12738, 41, 126jca31 524 . . . . . . . . . 10 (((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) ∧ (𝐹‘𝑛) = 𝑥) → ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)))
128127ex 418 . . . . . . . . 9 ((((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) ∧ 𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)) → ((𝐹‘𝑛) = 𝑥 → ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒))))
129128rexlimdva 3167 . . . . . . . 8 (((𝐹:𝑉⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑔‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥 → ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒))))
13026, 129syl3an1 1181 . . . . . . 7 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥 → ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒))))
13123, 130impbid 215 . . . . . 6 ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑋 ∈ 𝑉) → (((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)) ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
1321313exp 1137 . . . . 5 (𝐹 ∈ (𝐺 GraphIso 𝐻) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝑋 ∈ 𝑉 → (((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)) ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))))
133132impcom 413 . . . 4 (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝑋 ∈ 𝑉 → (((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)) ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥)))
134133imp 412 . . 3 ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)) ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
13516, 17clnbgrel 49141 . . . 4 (𝑥 ∈ (𝐻 ClNeighbVtx (𝐹‘𝑋)) ↔ ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒)))
136135a1i 11 . . 3 ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝐻 ClNeighbVtx (𝐹‘𝑋)) ↔ ((𝑥 ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑋) ∈ (Vtx‘𝐻)) ∧ (𝑥 = (𝐹‘𝑋) ∨ ∃𝑒 ∈ (Edg‘𝐻){(𝐹‘𝑋), 𝑥} ⊆ 𝑒))))
1372, 16grimf1o 49197 . . . . . 6 (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉⟶1-1onto(Vtx‘𝐻))
138137, 72syl 18 . . . . 5 (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹 Fn 𝑉)
139138adantl 487 . . . 4 (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 Fn 𝑉)
1402clnbgrssvtx 49144 . . . . 5 (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
141140a1i 11 . . . 4 (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉)
142 fvelimab 6983 . . . 4 ((𝐹 Fn 𝑉 ∧ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉) → (𝑥 ∈ (𝐹 “ (𝐺 ClNeighbVtx 𝑋)) ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
143139, 141, 142syl2an 608 . . 3 ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝐹 “ (𝐺 ClNeighbVtx 𝑋)) ↔ ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑥))
144134, 136, 1433bitr4d 314 . 2 ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝐻 ClNeighbVtx (𝐹‘𝑋)) ↔ 𝑥 ∈ (𝐹 “ (𝐺 ClNeighbVtx 𝑋))))
145144eqrdv 2761 1 ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝐻 ClNeighbVtx (𝐹‘𝑋)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑋)))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  {cpr 4590  dom cdm 5656   “ cima 5659  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  iEdgciedg 29608  Edgcedg 29658  UHGraphcuhgr 29667   ClNeighbVtx cclnbgr 49131   GraphIso cgrim 49188
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-edg 29659  df-uhgr 29669  df-clnbgr 49132  df-grim 49191
This theorem is used by:  uhgrimgrlim  49300
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