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Theorem ushgredgedg 26166
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 2651 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 ushgredgedg.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2ushgrf 26003 . . . 4 (𝐺 ∈ USHGraph → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
43adantr 480 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5 ssrab2 3720 . . 3 {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼
6 f1ores 6189 . . 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 695 . 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 2652 . . . . . 6 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ 𝑥𝐴) → (𝐼𝑥) = (𝐼𝑥))
1210, 11mpteq12dva 4765 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥𝐴 ↦ (𝐼𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
138, 12syl5eq 2697 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
14 f1f 6139 . . . . . . . 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 6289 . . . . . 6 (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
1817adantr 480 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
1918eqcomd 2657 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)) = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
2013, 19eqtrd 2685 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
21 ushgruhgr 26009 . . . . . . . . 9 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22 eqid 2651 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
2322uhgrfun 26006 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2421, 23syl 17 . . . . . . . 8 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
252funeqi 5947 . . . . . . . 8 (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
2624, 25sylibr 224 . . . . . . 7 (𝐺 ∈ USHGraph → Fun 𝐼)
2726adantr 480 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → Fun 𝐼)
28 dfimafn 6284 . . . . . 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 695 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒})
30 fveq2 6229 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
3130eleq2d 2716 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑁 ∈ (𝐼𝑖) ↔ 𝑁 ∈ (𝐼𝑗)))
3231elrab 3396 . . . . . . . . . 10 (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↔ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
33 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → 𝑗 ∈ dom 𝐼)
34 fvelrn 6392 . . . . . . . . . . . . . . . . 17 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran 𝐼)
352eqcomi 2660 . . . . . . . . . . . . . . . . . . 19 (iEdg‘𝐺) = 𝐼
3635rneqi 5384 . . . . . . . . . . . . . . . . . 18 ran (iEdg‘𝐺) = ran 𝐼
3736eleq2i 2722 . . . . . . . . . . . . . . . . 17 ((𝐼𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran 𝐼)
3834, 37sylibr 224 . . . . . . . . . . . . . . . 16 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
3927, 33, 38syl2an 493 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗))) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
40393adant3 1101 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
41 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐼𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4241eqcoms 2659 . . . . . . . . . . . . . . 15 ((𝐼𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
43423ad2ant3 1104 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4440, 43mpbird 247 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
45 ushgredgedg.e . . . . . . . . . . . . . . . . 17 𝐸 = (Edg‘𝐺)
46 edgval 25986 . . . . . . . . . . . . . . . . . 18 (Edg‘𝐺) = ran (iEdg‘𝐺)
4746a1i 11 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4845, 47syl5eq 2697 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
4948eleq2d 2716 . . . . . . . . . . . . . . 15 (𝐺 ∈ USHGraph → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5049adantr 480 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
51503ad2ant1 1102 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5244, 51mpbird 247 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓𝐸)
53 eleq2 2719 . . . . . . . . . . . . . . . 16 ((𝐼𝑗) = 𝑓 → (𝑁 ∈ (𝐼𝑗) ↔ 𝑁𝑓))
5453biimpcd 239 . . . . . . . . . . . . . . 15 (𝑁 ∈ (𝐼𝑗) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5554adantl 481 . . . . . . . . . . . . . 14 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5655a1i 11 . . . . . . . . . . . . 13 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓)))
57563imp 1275 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑁𝑓)
5852, 57jca 553 . . . . . . . . . . 11 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑁𝑓))
59583exp 1283 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6032, 59syl5bi 232 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6160rexlimdv 3059 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓)))
62 funfn 5956 . . . . . . . . . . . . . . 15 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
6362biimpi 206 . . . . . . . . . . . . . 14 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
6424, 63syl 17 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
65 fvelrnb 6282 . . . . . . . . . . . . 13 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6664, 65syl 17 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6735dmeqi 5357 . . . . . . . . . . . . . . . . . . . . . . 23 dom (iEdg‘𝐺) = dom 𝐼
6867eleq2i 2722 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
6968biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
7069adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
7170adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
7235fveq1i 6230 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((iEdg‘𝐺)‘𝑗) = (𝐼𝑗)
7372eqeq2i 2663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼𝑗))
7473biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼𝑗))
7574eqcoms 2659 . . . . . . . . . . . . . . . . . . . . . . . 24 (((iEdg‘𝐺)‘𝑗) = 𝑓𝑓 = (𝐼𝑗))
7675eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . 23 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓𝑁 ∈ (𝐼𝑗)))
7776biimpcd 239 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7877adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7978adantld 482 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼𝑗)))
8079imp 444 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼𝑗))
8171, 80jca 553 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
8281, 32sylibr 224 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})
8372eqeq1i 2656 . . . . . . . . . . . . . . . . . . . 20 (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼𝑗) = 𝑓)
8483biimpi 206 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = 𝑓)
8584adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼𝑗) = 𝑓)
8685adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼𝑗) = 𝑓)
8782, 86jca 553 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓))
8887ex 449 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓)))
8988reximdv2 3043 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9089ex 449 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (𝑁𝑓 → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9190com23 86 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9266, 91sylbid 230 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9349, 92sylbid 230 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (𝑓𝐸 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9493impd 446 . . . . . . . . 9 (𝐺 ∈ USHGraph → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9594adantr 480 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9661, 95impbid 202 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 ↔ (𝑓𝐸𝑁𝑓)))
97 vex 3234 . . . . . . . 8 𝑓 ∈ V
98 eqeq2 2662 . . . . . . . . 9 (𝑒 = 𝑓 → ((𝐼𝑗) = 𝑒 ↔ (𝐼𝑗) = 𝑓))
9998rexbidv 3081 . . . . . . . 8 (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
10097, 99elab 3382 . . . . . . 7 (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)
101 eleq2 2719 . . . . . . . 8 (𝑒 = 𝑓 → (𝑁𝑒𝑁𝑓))
102 ushgredgedg.b . . . . . . . 8 𝐵 = {𝑒𝐸𝑁𝑒}
103101, 102elrab2 3399 . . . . . . 7 (𝑓𝐵 ↔ (𝑓𝐸𝑁𝑓))
10496, 100, 1033bitr4g 303 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ 𝑓𝐵))
105104eqrdv 2649 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} = 𝐵)
10629, 105eqtrd 2685 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = 𝐵)
107106eqcomd 2657 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
10820, 10, 107f1oeq123d 6171 . 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)})))
1097, 108mpbird 247 1 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  {crab 2945  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  cmpt 4762  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UHGraphcuhgr 25996  USHGraphcushgr 25997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-edg 25985  df-uhgr 25998  df-ushgr 25999
This theorem is referenced by:  usgredgedg  26167  vtxdushgrfvedglem  26441
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