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Theorem ushgredgedg 29081
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 2725 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 ushgredgedg.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2ushgrf 28915 . . . 4 (𝐺 ∈ USHGraph → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
43adantr 479 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5 ssrab2 4070 . . 3 {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼
6 f1ores 6846 . . 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 584 . 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 2726 . . . . . 6 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ 𝑥𝐴) → (𝐼𝑥) = (𝐼𝑥))
1210, 11mpteq12dva 5233 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥𝐴 ↦ (𝐼𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
138, 12eqtrid 2777 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
14 f1f 6787 . . . . . . . 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 6961 . . . . . 6 (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
1817adantr 479 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
1918eqcomd 2731 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)) = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
2013, 19eqtrd 2765 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
21 ushgruhgr 28921 . . . . . . . . 9 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22 eqid 2725 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
2322uhgrfun 28918 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2421, 23syl 17 . . . . . . . 8 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
252funeqi 6569 . . . . . . . 8 (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
2624, 25sylibr 233 . . . . . . 7 (𝐺 ∈ USHGraph → Fun 𝐼)
2726adantr 479 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → Fun 𝐼)
28 dfimafn 6954 . . . . . 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 584 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒})
30 fveq2 6890 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
3130eleq2d 2811 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑁 ∈ (𝐼𝑖) ↔ 𝑁 ∈ (𝐼𝑗)))
3231elrab 3676 . . . . . . . . . 10 (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↔ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
33 simpl 481 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → 𝑗 ∈ dom 𝐼)
34 fvelrn 7079 . . . . . . . . . . . . . . . . 17 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran 𝐼)
352eqcomi 2734 . . . . . . . . . . . . . . . . . . 19 (iEdg‘𝐺) = 𝐼
3635rneqi 5934 . . . . . . . . . . . . . . . . . 18 ran (iEdg‘𝐺) = ran 𝐼
3736eleq2i 2817 . . . . . . . . . . . . . . . . 17 ((𝐼𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran 𝐼)
3834, 37sylibr 233 . . . . . . . . . . . . . . . 16 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
3927, 33, 38syl2an 594 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗))) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
40393adant3 1129 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
41 eleq1 2813 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐼𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4241eqcoms 2733 . . . . . . . . . . . . . . 15 ((𝐼𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
43423ad2ant3 1132 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4440, 43mpbird 256 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
45 ushgredgedg.e . . . . . . . . . . . . . . . . 17 𝐸 = (Edg‘𝐺)
46 edgval 28901 . . . . . . . . . . . . . . . . . 18 (Edg‘𝐺) = ran (iEdg‘𝐺)
4746a1i 11 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4845, 47eqtrid 2777 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
4948eleq2d 2811 . . . . . . . . . . . . . . 15 (𝐺 ∈ USHGraph → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5049adantr 479 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
51503ad2ant1 1130 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5244, 51mpbird 256 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓𝐸)
53 eleq2 2814 . . . . . . . . . . . . . . . 16 ((𝐼𝑗) = 𝑓 → (𝑁 ∈ (𝐼𝑗) ↔ 𝑁𝑓))
5453biimpcd 248 . . . . . . . . . . . . . . 15 (𝑁 ∈ (𝐼𝑗) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5554adantl 480 . . . . . . . . . . . . . 14 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5655a1i 11 . . . . . . . . . . . . 13 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓)))
57563imp 1108 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑁𝑓)
5852, 57jca 510 . . . . . . . . . . 11 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑁𝑓))
59583exp 1116 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6032, 59biimtrid 241 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6160rexlimdv 3143 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓)))
6224funfnd 6579 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63 fvelrnb 6952 . . . . . . . . . . . . 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 5902 . . . . . . . . . . . . . . . . . . . . . . 23 dom (iEdg‘𝐺) = dom 𝐼
6665eleq2i 2817 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
6766biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
6867adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
6968adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
7035fveq1i 6891 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((iEdg‘𝐺)‘𝑗) = (𝐼𝑗)
7170eqeq2i 2738 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼𝑗))
7271biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼𝑗))
7372eqcoms 2733 . . . . . . . . . . . . . . . . . . . . . . . 24 (((iEdg‘𝐺)‘𝑗) = 𝑓𝑓 = (𝐼𝑗))
7473eleq2d 2811 . . . . . . . . . . . . . . . . . . . . . . 23 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓𝑁 ∈ (𝐼𝑗)))
7574biimpcd 248 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7675adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7776adantld 489 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼𝑗)))
7877imp 405 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼𝑗))
7969, 78jca 510 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
8079, 32sylibr 233 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})
8170eqeq1i 2730 . . . . . . . . . . . . . . . . . . . 20 (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼𝑗) = 𝑓)
8281biimpi 215 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = 𝑓)
8382adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼𝑗) = 𝑓)
8483adantl 480 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼𝑗) = 𝑓)
8580, 84jca 510 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓))
8685ex 411 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓)))
8786reximdv2 3154 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
8887ex 411 . . . . . . . . . . . . 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 239 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9149, 90sylbid 239 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (𝑓𝐸 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9291impd 409 . . . . . . . . 9 (𝐺 ∈ USHGraph → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9392adantr 479 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9461, 93impbid 211 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 ↔ (𝑓𝐸𝑁𝑓)))
95 vex 3467 . . . . . . . 8 𝑓 ∈ V
96 eqeq2 2737 . . . . . . . . 9 (𝑒 = 𝑓 → ((𝐼𝑗) = 𝑒 ↔ (𝐼𝑗) = 𝑓))
9796rexbidv 3169 . . . . . . . 8 (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9895, 97elab 3661 . . . . . . 7 (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)
99 eleq2 2814 . . . . . . . 8 (𝑒 = 𝑓 → (𝑁𝑒𝑁𝑓))
100 ushgredgedg.b . . . . . . . 8 𝐵 = {𝑒𝐸𝑁𝑒}
10199, 100elrab2 3679 . . . . . . 7 (𝑓𝐵 ↔ (𝑓𝐸𝑁𝑓))
10294, 98, 1013bitr4g 313 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ 𝑓𝐵))
103102eqrdv 2723 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} = 𝐵)
10429, 103eqtrd 2765 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = 𝐵)
105104eqcomd 2731 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
10620, 10, 105f1oeq123d 6826 . 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 256 1 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2702  wrex 3060  {crab 3419  cdif 3938  wss 3941  c0 4319  𝒫 cpw 4599  {csn 4625  cmpt 5227  dom cdm 5673  ran crn 5674  cres 5675  cima 5676  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1wf1 6540  1-1-ontowf1o 6542  cfv 6543  Vtxcvtx 28848  iEdgciedg 28849  Edgcedg 28899  UHGraphcuhgr 28908  USHGraphcushgr 28909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-edg 28900  df-uhgr 28910  df-ushgr 28911
This theorem is referenced by:  usgredgedg  29082  vtxdushgrfvedglem  29342
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