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Theorem uspgrlimlem4 47931
Description: Lemma 4 for uspgrlim 47932. (Contributed by AV, 16-Aug-2025.)
Hypotheses
Ref Expression
uspgrlim.v 𝑉 = (Vtx‘𝐺)
uspgrlim.w 𝑊 = (Vtx‘𝐻)
uspgrlim.n 𝑁 = (𝐺 ClNeighbVtx 𝑣)
uspgrlim.m 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
uspgrlim.i 𝐼 = (Edg‘𝐺)
uspgrlim.j 𝐽 = (Edg‘𝐻)
uspgrlim.k 𝐾 = {𝑥𝐼𝑥𝑁}
uspgrlim.l 𝐿 = {𝑥𝐽𝑥𝑀}
Assertion
Ref Expression
uspgrlimlem4 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))
Distinct variable groups:   𝑖,𝐺,𝑥   𝑖,𝐻,𝑥   𝑥,𝐼   𝑥,𝐽   𝑥,𝑀   𝑖,𝑁,𝑥   𝑒,𝑖,𝑥   𝑓,𝑖   𝑒,𝐺   𝑒,𝐾,𝑥   𝑥,𝐿   𝑒,𝑓   𝑒,𝑔
Allowed substitution hints:   𝐹(𝑥,𝑣,𝑒,𝑓,𝑔,𝑖)   𝐺(𝑣,𝑓,𝑔)   𝐻(𝑣,𝑒,𝑓,𝑔)   𝐼(𝑣,𝑒,𝑓,𝑔,𝑖)   𝐽(𝑣,𝑒,𝑓,𝑔,𝑖)   𝐾(𝑣,𝑓,𝑔,𝑖)   𝐿(𝑣,𝑒,𝑓,𝑔,𝑖)   𝑀(𝑣,𝑒,𝑓,𝑔,𝑖)   𝑁(𝑣,𝑒,𝑓,𝑔)   𝑉(𝑥,𝑣,𝑒,𝑓,𝑔,𝑖)   𝑊(𝑥,𝑣,𝑒,𝑓,𝑔,𝑖)

Proof of Theorem uspgrlimlem4
StepHypRef Expression
1 eqid 2736 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
21uspgrf1oedg 29180 . . . . . 6 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
3 f1of 6846 . . . . . 6 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
42, 3syl 17 . . . . 5 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
54ad2antrr 726 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
6 simpl 482 . . . 4 ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → 𝑖 ∈ dom (iEdg‘𝐺))
7 fvco3 7006 . . . . 5 (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖) = (((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖)))
87fveq2d 6908 . . . 4 (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐻)‘(((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))))
95, 6, 8syl2an 596 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐻)‘(((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))))
10 eqid 2736 . . . . . . 7 (iEdg‘𝐻) = (iEdg‘𝐻)
1110uspgrf1oedg 29180 . . . . . 6 (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
1211ad3antlr 731 . . . . 5 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
13 ssrab2 4079 . . . . . . 7 {𝑥𝐽𝑥𝑀} ⊆ 𝐽
14 uspgrlim.l . . . . . . 7 𝐿 = {𝑥𝐽𝑥𝑀}
15 uspgrlim.j . . . . . . . 8 𝐽 = (Edg‘𝐻)
1615eqcomi 2745 . . . . . . 7 (Edg‘𝐻) = 𝐽
1713, 14, 163sstr4i 4034 . . . . . 6 𝐿 ⊆ (Edg‘𝐻)
18 f1of 6846 . . . . . . . . . 10 (𝑔:𝐾1-1-onto𝐿𝑔:𝐾𝐿)
1918adantr 480 . . . . . . . . 9 ((𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒)) → 𝑔:𝐾𝐿)
2019adantl 481 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) → 𝑔:𝐾𝐿)
2120adantr 480 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → 𝑔:𝐾𝐿)
225ffund 6738 . . . . . . . . . 10 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) → Fun (iEdg‘𝐺))
231iedgedg 29057 . . . . . . . . . 10 ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ (Edg‘𝐺))
2422, 6, 23syl2an 596 . . . . . . . . 9 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ (Edg‘𝐺))
25 uspgrlim.i . . . . . . . . 9 𝐼 = (Edg‘𝐺)
2624, 25eleqtrrdi 2851 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐼)
27 simprr 773 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)
28 sseq1 4008 . . . . . . . . 9 (𝑥 = ((iEdg‘𝐺)‘𝑖) → (𝑥𝑁 ↔ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁))
29 uspgrlim.k . . . . . . . . 9 𝐾 = {𝑥𝐼𝑥𝑁}
3028, 29elrab2 3694 . . . . . . . 8 (((iEdg‘𝐺)‘𝑖) ∈ 𝐾 ↔ (((iEdg‘𝐺)‘𝑖) ∈ 𝐼 ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁))
3126, 27, 30sylanbrc 583 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐾)
3221, 31ffvelcdmd 7103 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑔‘((iEdg‘𝐺)‘𝑖)) ∈ 𝐿)
3317, 32sselid 3980 . . . . 5 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑔‘((iEdg‘𝐺)‘𝑖)) ∈ (Edg‘𝐻))
34 f1ocnvfv2 7295 . . . . 5 (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ (𝑔‘((iEdg‘𝐺)‘𝑖)) ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
3512, 33, 34syl2anc 584 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘((iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
36 fvco3 7006 . . . . . 6 ((𝑔:𝐾𝐿 ∧ ((iEdg‘𝐺)‘𝑖) ∈ 𝐾) → (((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖))))
3736fveq2d 6908 . . . . 5 ((𝑔:𝐾𝐿 ∧ ((iEdg‘𝐺)‘𝑖) ∈ 𝐾) → ((iEdg‘𝐻)‘(((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))) = ((iEdg‘𝐻)‘((iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))))
3821, 31, 37syl2anc 584 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘(((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))) = ((iEdg‘𝐻)‘((iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))))
3925eqcomi 2745 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = 𝐼
40 feq3 6716 . . . . . . . . . . . . . . 15 ((Edg‘𝐺) = 𝐼 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼))
4139, 40ax-mp 5 . . . . . . . . . . . . . 14 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
4241biimpi 216 . . . . . . . . . . . . 13 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
432, 3, 423syl 18 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
4443ad2antrr 726 . . . . . . . . . . 11 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
456adantl 481 . . . . . . . . . . 11 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → 𝑖 ∈ dom (iEdg‘𝐺))
4644, 45ffvelcdmd 7103 . . . . . . . . . 10 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐼)
47 simprr 773 . . . . . . . . . 10 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)
4846, 47, 30sylanbrc 583 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐾)
49 imaeq2 6072 . . . . . . . . . . 11 (𝑒 = ((iEdg‘𝐺)‘𝑖) → (𝑓𝑒) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))
50 fveq2 6904 . . . . . . . . . . 11 (𝑒 = ((iEdg‘𝐺)‘𝑖) → (𝑔𝑒) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
5149, 50eqeq12d 2752 . . . . . . . . . 10 (𝑒 = ((iEdg‘𝐺)‘𝑖) → ((𝑓𝑒) = (𝑔𝑒) ↔ (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
5251rspcv 3617 . . . . . . . . 9 (((iEdg‘𝐺)‘𝑖) ∈ 𝐾 → (∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
5348, 52syl 17 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
5453ex 412 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))))
5554com23 86 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))))
5655adantld 490 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒)) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))))
5756imp31 417 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
5835, 38, 573eqtr4d 2786 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘(((iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))
599, 58eqtr2d 2777 . 2 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖)))
6059ex 412 1 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3060  {crab 3435  wss 3950  ccnv 5682  dom cdm 5683  cima 5686  ccom 5687  Fun wfun 6553  wf 6555  1-1-ontowf1o 6558  cfv 6559  (class class class)co 7429  Vtxcvtx 29003  iEdgciedg 29004  Edgcedg 29054  USPGraphcuspgr 29155   ClNeighbVtx cclnbgr 47778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-edg 29055  df-uspgr 29157
This theorem is referenced by:  uspgrlim  47932
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