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Theorem ushgredgedg 27317
Description: In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.)
Hypotheses
Ref Expression
ushgredgedg.e 𝐸 = (Edg‘𝐺)
ushgredgedg.i 𝐼 = (iEdg‘𝐺)
ushgredgedg.v 𝑉 = (Vtx‘𝐺)
ushgredgedg.a 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
ushgredgedg.b 𝐵 = {𝑒𝐸𝑁𝑒}
ushgredgedg.f 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
Assertion
Ref Expression
ushgredgedg ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐵,𝑒   𝑒,𝐸,𝑖   𝑒,𝐺,𝑖,𝑥   𝑒,𝐼,𝑖,𝑥   𝑒,𝑁,𝑖,𝑥   𝑒,𝑉,𝑖,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑒,𝑖)   𝐵(𝑥,𝑖)   𝐸(𝑥)   𝐹(𝑥,𝑒,𝑖)

Proof of Theorem ushgredgedg
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 ushgredgedg.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2ushgrf 27154 . . . 4 (𝐺 ∈ USHGraph → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
43adantr 484 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5 ssrab2 3993 . . 3 {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼
6 f1ores 6675 . . 3 ((𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼) → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}):{𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
74, 5, 6sylancl 589 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}):{𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
8 ushgredgedg.f . . . . 5 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
9 ushgredgedg.a . . . . . . 7 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
109a1i 11 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})
11 eqidd 2738 . . . . . 6 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ 𝑥𝐴) → (𝐼𝑥) = (𝐼𝑥))
1210, 11mpteq12dva 5139 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥𝐴 ↦ (𝐼𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
138, 12syl5eq 2790 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
14 f1f 6615 . . . . . . . 8 (𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
153, 14syl 17 . . . . . . 7 (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
165a1i 11 . . . . . . 7 (𝐺 ∈ USHGraph → {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼)
1715, 16feqresmpt 6781 . . . . . 6 (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
1817adantr 484 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
1918eqcomd 2743 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)) = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
2013, 19eqtrd 2777 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
21 ushgruhgr 27160 . . . . . . . . 9 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22 eqid 2737 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
2322uhgrfun 27157 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2421, 23syl 17 . . . . . . . 8 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
252funeqi 6401 . . . . . . . 8 (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
2624, 25sylibr 237 . . . . . . 7 (𝐺 ∈ USHGraph → Fun 𝐼)
2726adantr 484 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → Fun 𝐼)
28 dfimafn 6775 . . . . . 6 ((Fun 𝐼 ∧ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒})
2927, 5, 28sylancl 589 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒})
30 fveq2 6717 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
3130eleq2d 2823 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑁 ∈ (𝐼𝑖) ↔ 𝑁 ∈ (𝐼𝑗)))
3231elrab 3602 . . . . . . . . . 10 (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↔ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
33 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → 𝑗 ∈ dom 𝐼)
34 fvelrn 6897 . . . . . . . . . . . . . . . . 17 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran 𝐼)
352eqcomi 2746 . . . . . . . . . . . . . . . . . . 19 (iEdg‘𝐺) = 𝐼
3635rneqi 5806 . . . . . . . . . . . . . . . . . 18 ran (iEdg‘𝐺) = ran 𝐼
3736eleq2i 2829 . . . . . . . . . . . . . . . . 17 ((𝐼𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran 𝐼)
3834, 37sylibr 237 . . . . . . . . . . . . . . . 16 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
3927, 33, 38syl2an 599 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗))) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
40393adant3 1134 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
41 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐼𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4241eqcoms 2745 . . . . . . . . . . . . . . 15 ((𝐼𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
43423ad2ant3 1137 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4440, 43mpbird 260 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
45 ushgredgedg.e . . . . . . . . . . . . . . . . 17 𝐸 = (Edg‘𝐺)
46 edgval 27140 . . . . . . . . . . . . . . . . . 18 (Edg‘𝐺) = ran (iEdg‘𝐺)
4746a1i 11 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4845, 47syl5eq 2790 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
4948eleq2d 2823 . . . . . . . . . . . . . . 15 (𝐺 ∈ USHGraph → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5049adantr 484 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
51503ad2ant1 1135 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5244, 51mpbird 260 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓𝐸)
53 eleq2 2826 . . . . . . . . . . . . . . . 16 ((𝐼𝑗) = 𝑓 → (𝑁 ∈ (𝐼𝑗) ↔ 𝑁𝑓))
5453biimpcd 252 . . . . . . . . . . . . . . 15 (𝑁 ∈ (𝐼𝑗) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5554adantl 485 . . . . . . . . . . . . . 14 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5655a1i 11 . . . . . . . . . . . . 13 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓)))
57563imp 1113 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑁𝑓)
5852, 57jca 515 . . . . . . . . . . 11 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑁𝑓))
59583exp 1121 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6032, 59syl5bi 245 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6160rexlimdv 3202 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓)))
6224funfnd 6411 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63 fvelrnb 6773 . . . . . . . . . . . . 13 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6462, 63syl 17 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6535dmeqi 5773 . . . . . . . . . . . . . . . . . . . . . . 23 dom (iEdg‘𝐺) = dom 𝐼
6665eleq2i 2829 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
6766biimpi 219 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
6867adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
6968adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
7035fveq1i 6718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((iEdg‘𝐺)‘𝑗) = (𝐼𝑗)
7170eqeq2i 2750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼𝑗))
7271biimpi 219 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼𝑗))
7372eqcoms 2745 . . . . . . . . . . . . . . . . . . . . . . . 24 (((iEdg‘𝐺)‘𝑗) = 𝑓𝑓 = (𝐼𝑗))
7473eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . 23 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓𝑁 ∈ (𝐼𝑗)))
7574biimpcd 252 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7675adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7776adantld 494 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼𝑗)))
7877imp 410 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼𝑗))
7969, 78jca 515 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
8079, 32sylibr 237 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})
8170eqeq1i 2742 . . . . . . . . . . . . . . . . . . . 20 (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼𝑗) = 𝑓)
8281biimpi 219 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = 𝑓)
8382adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼𝑗) = 𝑓)
8483adantl 485 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼𝑗) = 𝑓)
8580, 84jca 515 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓))
8685ex 416 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓)))
8786reximdv2 3190 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
8887ex 416 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (𝑁𝑓 → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
8988com23 86 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9064, 89sylbid 243 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9149, 90sylbid 243 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (𝑓𝐸 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9291impd 414 . . . . . . . . 9 (𝐺 ∈ USHGraph → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9392adantr 484 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9461, 93impbid 215 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 ↔ (𝑓𝐸𝑁𝑓)))
95 vex 3412 . . . . . . . 8 𝑓 ∈ V
96 eqeq2 2749 . . . . . . . . 9 (𝑒 = 𝑓 → ((𝐼𝑗) = 𝑒 ↔ (𝐼𝑗) = 𝑓))
9796rexbidv 3216 . . . . . . . 8 (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9895, 97elab 3587 . . . . . . 7 (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)
99 eleq2 2826 . . . . . . . 8 (𝑒 = 𝑓 → (𝑁𝑒𝑁𝑓))
100 ushgredgedg.b . . . . . . . 8 𝐵 = {𝑒𝐸𝑁𝑒}
10199, 100elrab2 3605 . . . . . . 7 (𝑓𝐵 ↔ (𝑓𝐸𝑁𝑓))
10294, 98, 1013bitr4g 317 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ 𝑓𝐵))
103102eqrdv 2735 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} = 𝐵)
10429, 103eqtrd 2777 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = 𝐵)
105104eqcomd 2743 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
10620, 10, 105f1oeq123d 6655 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐹:𝐴1-1-onto𝐵 ↔ (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}):{𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})))
1077, 106mpbird 260 1 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  {cab 2714  wrex 3062  {crab 3065  cdif 3863  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541  cmpt 5135  dom cdm 5551  ran crn 5552  cres 5553  cima 5554  Fun wfun 6374   Fn wfn 6375  wf 6376  1-1wf1 6377  1-1-ontowf1o 6379  cfv 6380  Vtxcvtx 27087  iEdgciedg 27088  Edgcedg 27138  UHGraphcuhgr 27147  USHGraphcushgr 27148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-edg 27139  df-uhgr 27149  df-ushgr 27150
This theorem is referenced by:  usgredgedg  27318  vtxdushgrfvedglem  27577
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