Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ushgredgedga Structured version   Visualization version   GIF version

Theorem ushgredgedga 40456
 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
ushgredgedga.e 𝐸 = (Edg‘𝐺)
ushgredgedga.i 𝐼 = (iEdg‘𝐺)
ushgredgedga.v 𝑉 = (Vtx‘𝐺)
ushgredgedga.a 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
ushgredgedga.b 𝐵 = {𝑒𝐸𝑁𝑒}
ushgredgedga.f 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
Assertion
Ref Expression
ushgredgedga ((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 ushgredgedga
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 ushgredgedga.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2ushgrf 25729 . . . 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 3650 . . 3 {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ⊆ dom 𝐼
6 f1ores 6064 . . 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 693 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}):{𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
8 ushgredgedga.f . . . . 5 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
9 ushgredgedga.a . . . . . . 7 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
109a1i 11 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})
11 eqidd 2611 . . . . . 6 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ 𝑥𝐴) → (𝐼𝑥) = (𝐼𝑥))
1210, 11mpteq12dva 4662 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥𝐴 ↦ (𝐼𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
138, 12syl5eq 2656 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)))
14 f1f 6014 . . . . . . . 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 6160 . . . . . 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 2616 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↦ (𝐼𝑥)) = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
2013, 19eqtrd 2644 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
21 ushgruhgr 25735 . . . . . . . . 9 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )
22 eqid 2610 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
2322uhgrfun 25732 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2421, 23syl 17 . . . . . . . 8 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
252funeqi 5824 . . . . . . . 8 (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
2624, 25sylibr 223 . . . . . . 7 (𝐺 ∈ USHGraph → Fun 𝐼)
2726adantr 480 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → Fun 𝐼)
28 dfimafn 6155 . . . . . 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 693 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒})
30 fveq2 6103 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
3130eleq2d 2673 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑁 ∈ (𝐼𝑖) ↔ 𝑁 ∈ (𝐼𝑗)))
3231elrab 3331 . . . . . . . . . 10 (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ↔ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
33 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → 𝑗 ∈ dom 𝐼)
34 fvelrn 6260 . . . . . . . . . . . . . . . . 17 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran 𝐼)
352eqcomi 2619 . . . . . . . . . . . . . . . . . . 19 (iEdg‘𝐺) = 𝐼
3635rneqi 5273 . . . . . . . . . . . . . . . . . 18 ran (iEdg‘𝐺) = ran 𝐼
3736eleq2i 2680 . . . . . . . . . . . . . . . . 17 ((𝐼𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran 𝐼)
3834, 37sylibr 223 . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
41 eleq1 2676 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐼𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4241eqcoms 2618 . . . . . . . . . . . . . . 15 ((𝐼𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
43423ad2ant3 1077 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4440, 43mpbird 246 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
45 ushgredgedga.e . . . . . . . . . . . . . . . . 17 𝐸 = (Edg‘𝐺)
46 edgaval 25794 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4745, 46syl5eq 2656 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
4847eleq2d 2673 . . . . . . . . . . . . . . 15 (𝐺 ∈ USHGraph → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
4948adantr 480 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
50493ad2ant1 1075 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5144, 50mpbird 246 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑓𝐸)
52 eleq2 2677 . . . . . . . . . . . . . . . 16 ((𝐼𝑗) = 𝑓 → (𝑁 ∈ (𝐼𝑗) ↔ 𝑁𝑓))
5352biimpcd 238 . . . . . . . . . . . . . . 15 (𝑁 ∈ (𝐼𝑗) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5453adantl 481 . . . . . . . . . . . . . 14 ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓))
5554a1i 11 . . . . . . . . . . . . 13 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓𝑁𝑓)))
56553imp 1249 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → 𝑁𝑓)
5751, 56jca 553 . . . . . . . . . . 11 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑁𝑓))
58573exp 1256 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)) → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
5932, 58syl5bi 231 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓))))
6059rexlimdv 3012 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 → (𝑓𝐸𝑁𝑓)))
61 funfn 5833 . . . . . . . . . . . . . . 15 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
6261biimpi 205 . . . . . . . . . . . . . 14 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
6324, 62syl 17 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
64 fvelrnb 6153 . . . . . . . . . . . . 13 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6563, 64syl 17 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6635dmeqi 5247 . . . . . . . . . . . . . . . . . . . . . . 23 dom (iEdg‘𝐺) = dom 𝐼
6766eleq2i 2680 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
6867biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
6968adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
7069adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
7135fveq1i 6104 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((iEdg‘𝐺)‘𝑗) = (𝐼𝑗)
7271eqeq2i 2622 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼𝑗))
7372biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼𝑗))
7473eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (((iEdg‘𝐺)‘𝑗) = 𝑓𝑓 = (𝐼𝑗))
7574eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . 23 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓𝑁 ∈ (𝐼𝑗)))
7675biimpcd 238 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7776adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓𝑁 ∈ (𝐼𝑗)))
7877adantld 482 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼𝑗)))
7978imp 444 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼𝑗))
8070, 79jca 553 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼𝑁 ∈ (𝐼𝑗)))
8180, 32sylibr 223 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)})
8271eqeq1i 2615 . . . . . . . . . . . . . . . . . . . 20 (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼𝑗) = 𝑓)
8382biimpi 205 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = 𝑓)
8483adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼𝑗) = 𝑓)
8584adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼𝑗) = 𝑓)
8681, 85jca 553 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USHGraph ∧ 𝑁𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓))
8786ex 449 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} ∧ (𝐼𝑗) = 𝑓)))
8887reximdv2 2997 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑁𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
8988ex 449 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → (𝑁𝑓 → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9089com23 84 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9165, 90sylbid 229 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9248, 91sylbid 229 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (𝑓𝐸 → (𝑁𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)))
9392impd 446 . . . . . . . . 9 (𝐺 ∈ USHGraph → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9493adantr 480 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑓𝐸𝑁𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9560, 94impbid 201 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓 ↔ (𝑓𝐸𝑁𝑓)))
96 vex 3176 . . . . . . . 8 𝑓 ∈ V
97 eqeq2 2621 . . . . . . . . 9 (𝑒 = 𝑓 → ((𝐼𝑗) = 𝑒 ↔ (𝐼𝑗) = 𝑓))
9897rexbidv 3034 . . . . . . . 8 (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓))
9996, 98elab 3319 . . . . . . 7 (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑓)
100 eleq2 2677 . . . . . . . 8 (𝑒 = 𝑓 → (𝑁𝑒𝑁𝑓))
101 ushgredgedga.b . . . . . . . 8 𝐵 = {𝑒𝐸𝑁𝑒}
102100, 101elrab2 3333 . . . . . . 7 (𝑓𝐵 ↔ (𝑓𝐸𝑁𝑓))
10395, 99, 1023bitr4g 302 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} ↔ 𝑓𝐵))
104103eqrdv 2608 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)} (𝐼𝑗) = 𝑒} = 𝐵)
10529, 104eqtrd 2644 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}) = 𝐵)
106105eqcomd 2616 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}))
10720, 10, 106f1oeq123d 6046 . 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)})))
1087, 107mpbird 246 1 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   USHGraph cushgr 25723  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-uhgr 25724  df-ushgr 25725  df-edga 25793 This theorem is referenced by:  usgredgedga  40457  vtxdushgrfvedglem  40704
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