Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ushgredgedg Structured version   Visualization version   GIF version

Theorem ushgredgedg 27026
 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 2798 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 ushgredgedg.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2ushgrf 26863 . . . 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 4007 . . 3 {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼
6 f1ores 6604 . . 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 2799 . . . . . 6 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ 𝑥𝐴) → (𝐼𝑥) = (𝐼𝑥))
1210, 11mpteq12dva 5114 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥𝐴 ↦ (𝐼𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
138, 12syl5eq 2845 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
14 f1f 6549 . . . . . . . 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 6709 . . . . . 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 2804 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)) = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
2013, 19eqtrd 2833 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
21 ushgruhgr 26869 . . . . . . . . 9 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22 eqid 2798 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
2322uhgrfun 26866 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2421, 23syl 17 . . . . . . . 8 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
252funeqi 6345 . . . . . . . 8 (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
2624, 25sylibr 237 . . . . . . 7 (𝐺 ∈ USHGraph → Fun 𝐼)
2726adantr 484 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → Fun 𝐼)
28 dfimafn 6703 . . . . . 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 6645 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
3130eleq2d 2875 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑁 ∈ (𝐼𝑖) ↔ 𝑁 ∈ (𝐼𝑗)))
3231elrab 3628 . . . . . . . . . 10 (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↔ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
33 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → 𝑗 ∈ dom 𝐼)
34 fvelrn 6821 . . . . . . . . . . . . . . . . 17 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran 𝐼)
352eqcomi 2807 . . . . . . . . . . . . . . . . . . 19 (iEdg‘𝐺) = 𝐼
3635rneqi 5771 . . . . . . . . . . . . . . . . . 18 ran (iEdg‘𝐺) = ran 𝐼
3736eleq2i 2881 . . . . . . . . . . . . . . . . 17 ((𝐼𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran 𝐼)
3834, 37sylibr 237 . . . . . . . . . . . . . . . 16 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
3927, 33, 38syl2an 598 . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐼𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4241eqcoms 2806 . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
45 ushgredgedg.e . . . . . . . . . . . . . . . . 17 𝐸 = (Edg‘𝐺)
46 edgval 26849 . . . . . . . . . . . . . . . . . 18 (Edg‘𝐺) = ran (iEdg‘𝐺)
4746a1i 11 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4845, 47syl5eq 2845 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
4948eleq2d 2875 . . . . . . . . . . . . . . 15 (𝐺 ∈ USHGraph → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5049adantr 484 . . . . . . . . . . . . . 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 260 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓𝐸)
53 eleq2 2878 . . . . . . . . . . . . . . . 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 1108 . . . . . . . . . . . 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 1116 . . . . . . . . . 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 3242 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓)))
6224funfnd 6355 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63 fvelrnb 6701 . . . . . . . . . . . . 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 5737 . . . . . . . . . . . . . . . . . . . . . . 23 dom (iEdg‘𝐺) = dom 𝐼
6665eleq2i 2881 . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((iEdg‘𝐺)‘𝑗) = (𝐼𝑗)
7170eqeq2i 2811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼𝑗))
7271biimpi 219 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼𝑗))
7372eqcoms 2806 . . . . . . . . . . . . . . . . . . . . . . . 24 (((iEdg‘𝐺)‘𝑗) = 𝑓𝑓 = (𝐼𝑗))
7473eleq2d 2875 . . . . . . . . . . . . . . . . . . . . . . 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 2803 . . . . . . . . . . . . . . . . . . . 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 3230 . . . . . . . . . . . . . 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 3444 . . . . . . . 8 𝑓 ∈ V
96 eqeq2 2810 . . . . . . . . 9 (𝑒 = 𝑓 → ((𝐼𝑗) = 𝑒 ↔ (𝐼𝑗) = 𝑓))
9796rexbidv 3256 . . . . . . . 8 (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9895, 97elab 3615 . . . . . . 7 (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)
99 eleq2 2878 . . . . . . . 8 (𝑒 = 𝑓 → (𝑁𝑒𝑁𝑓))
100 ushgredgedg.b . . . . . . . 8 𝐵 = {𝑒𝐸𝑁𝑒}
10199, 100elrab2 3631 . . . . . . 7 (𝑓𝐵 ↔ (𝑓𝐸𝑁𝑓))
10294, 98, 1013bitr4g 317 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ 𝑓𝐵))
103102eqrdv 2796 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} = 𝐵)
10429, 103eqtrd 2833 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = 𝐵)
105104eqcomd 2804 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
10620, 10, 105f1oeq123d 6585 . 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 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525   ↦ cmpt 5110  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  –1-1→wf1 6321  –1-1-onto→wf1o 6323  ‘cfv 6324  Vtxcvtx 26796  iEdgciedg 26797  Edgcedg 26847  UHGraphcuhgr 26856  USHGraphcushgr 26857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-edg 26848  df-uhgr 26858  df-ushgr 26859 This theorem is referenced by:  usgredgedg  27027  vtxdushgrfvedglem  27286
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