Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isomuspgrlem1 Structured version   Visualization version   GIF version

Theorem isomuspgrlem1 42744
 Description: Lemma 1 for isomuspgr 42751. (Contributed by AV, 29-Nov-2022.)
Hypotheses
Ref Expression
isomushgr.v 𝑉 = (Vtx‘𝐴)
isomushgr.w 𝑊 = (Vtx‘𝐵)
isomushgr.e 𝐸 = (Edg‘𝐴)
isomushgr.k 𝐾 = (Edg‘𝐵)
Assertion
Ref Expression
isomuspgrlem1 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ({(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))
Distinct variable groups:   𝐴,𝑒,𝑓,𝑔   𝐵,𝑒,𝑓,𝑔   𝑒,𝐸,𝑔   𝑔,𝐾   𝑒,𝑉,𝑔   𝑒,𝑊,𝑔   𝑎,𝑏,𝑔,𝑓
Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐸(𝑓,𝑎,𝑏)   𝐾(𝑒,𝑓,𝑎,𝑏)   𝑉(𝑓,𝑎,𝑏)   𝑊(𝑓,𝑎,𝑏)

Proof of Theorem isomuspgrlem1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 476 . . . . 5 ((𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)) → 𝑔:𝐸1-1-onto𝐾)
21ad2antlr 717 . . . 4 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → 𝑔:𝐸1-1-onto𝐾)
3 f1ocnvdm 6812 . . . 4 ((𝑔:𝐸1-1-onto𝐾 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸)
42, 3sylan 575 . . 3 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸)
5 uspgrupgr 26525 . . . . . . 7 (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
65adantr 474 . . . . . 6 ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
76ad4antr 722 . . . . 5 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → 𝐴 ∈ UPGraph)
8 isomushgr.v . . . . . 6 𝑉 = (Vtx‘𝐴)
9 isomushgr.e . . . . . 6 𝐸 = (Edg‘𝐴)
108, 9upgredg 26486 . . . . 5 ((𝐴 ∈ UPGraph ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦})
117, 10sylan 575 . . . 4 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦})
12 eleq1 2847 . . . . . . . . . . 11 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
1312biimpd 221 . . . . . . . . . 10 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 → {𝑥, 𝑦} ∈ 𝐸))
1413com12 32 . . . . . . . . 9 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑥, 𝑦} ∈ 𝐸))
1514ad2antlr 717 . . . . . . . 8 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑥, 𝑦} ∈ 𝐸))
1615imp 397 . . . . . . 7 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}) → {𝑥, 𝑦} ∈ 𝐸)
171ad6antlr 727 . . . . . . . . . . 11 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑔:𝐸1-1-onto𝐾)
18 simpr 479 . . . . . . . . . . 11 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → {𝑥, 𝑦} ∈ 𝐸)
19 simpr 479 . . . . . . . . . . . 12 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾)
2019ad5ant12 746 . . . . . . . . . . 11 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾)
2117, 18, 203jca 1119 . . . . . . . . . 10 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑔:𝐸1-1-onto𝐾 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾))
22 f1ocnvfvb 6807 . . . . . . . . . 10 ((𝑔:𝐸1-1-onto𝐾 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}))
2321, 22syl 17 . . . . . . . . 9 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}))
24 imaeq2 5716 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑓𝑒) = (𝑓 “ {𝑥, 𝑦}))
25 fveq2 6446 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑔𝑒) = (𝑔‘{𝑥, 𝑦}))
2624, 25eqeq12d 2793 . . . . . . . . . . . . . . 15 (𝑒 = {𝑥, 𝑦} → ((𝑓𝑒) = (𝑔𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2726rspccv 3508 . . . . . . . . . . . . . 14 (∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2827adantl 475 . . . . . . . . . . . . 13 ((𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2928adantl 475 . . . . . . . . . . . 12 ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
3029ad4antr 722 . . . . . . . . . . 11 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
3130imp 397 . . . . . . . . . 10 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}))
32 eqeq1 2782 . . . . . . . . . . . . 13 ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3332eqcoms 2786 . . . . . . . . . . . 12 ((𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3433adantl 475 . . . . . . . . . . 11 ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
35 f1ofn 6392 . . . . . . . . . . . . . . . . . 18 (𝑓:𝑉1-1-onto𝑊𝑓 Fn 𝑉)
3635ad6antlr 727 . . . . . . . . . . . . . . . . 17 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → 𝑓 Fn 𝑉)
37 simpl 476 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑦𝑉) → 𝑥𝑉)
3837adantl 475 . . . . . . . . . . . . . . . . 17 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
39 simpr 479 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑦𝑉) → 𝑦𝑉)
4039adantl 475 . . . . . . . . . . . . . . . . 17 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
4136, 38, 403jca 1119 . . . . . . . . . . . . . . . 16 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → (𝑓 Fn 𝑉𝑥𝑉𝑦𝑉))
4241adantr 474 . . . . . . . . . . . . . . 15 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 Fn 𝑉𝑥𝑉𝑦𝑉))
43 fnimapr 6522 . . . . . . . . . . . . . . 15 ((𝑓 Fn 𝑉𝑥𝑉𝑦𝑉) → (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑥), (𝑓𝑦)})
4442, 43syl 17 . . . . . . . . . . . . . 14 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑥), (𝑓𝑦)})
4544eqeq1d 2780 . . . . . . . . . . . . 13 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}))
46 fvex 6459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑥) ∈ V
47 fvex 6459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑦) ∈ V
48 fvex 6459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑎) ∈ V
49 fvex 6459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑏) ∈ V
5046, 47, 48, 49preq12b 4610 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} ↔ (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))))
51 f1of1 6390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:𝑉1-1-onto𝑊𝑓:𝑉1-1𝑊)
52 simpl 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑎𝑉)
5352, 37anim12ci 607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑎𝑉))
54 f1veqaeq 6786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑥𝑉𝑎𝑉)) → ((𝑓𝑥) = (𝑓𝑎) → 𝑥 = 𝑎))
5551, 53, 54syl2anr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑥) = (𝑓𝑎) → 𝑥 = 𝑎))
56 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑏𝑉)
5756, 39anim12ci 607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑦𝑉𝑏𝑉))
58 f1veqaeq 6786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑦𝑉𝑏𝑉)) → ((𝑓𝑦) = (𝑓𝑏) → 𝑦 = 𝑏))
5951, 57, 58syl2anr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑦) = (𝑓𝑏) → 𝑦 = 𝑏))
6055, 59anim12d 602 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) → (𝑥 = 𝑎𝑦 = 𝑏)))
6160impcom 398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → (𝑥 = 𝑎𝑦 = 𝑏))
6261orcd 862 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
6362ex 403 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
6456, 37anim12ci 607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑏𝑉))
65 f1veqaeq 6786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑥𝑉𝑏𝑉)) → ((𝑓𝑥) = (𝑓𝑏) → 𝑥 = 𝑏))
6651, 64, 65syl2anr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑥) = (𝑓𝑏) → 𝑥 = 𝑏))
6752, 39anim12ci 607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑦𝑉𝑎𝑉))
68 f1veqaeq 6786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑦𝑉𝑎𝑉)) → ((𝑓𝑦) = (𝑓𝑎) → 𝑦 = 𝑎))
6951, 67, 68syl2anr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑦) = (𝑓𝑎) → 𝑦 = 𝑎))
7066, 69anim12d 602 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → (((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) → (𝑥 = 𝑏𝑦 = 𝑎)))
7170impcom 398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → (𝑥 = 𝑏𝑦 = 𝑎))
7271olcd 863 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
7372ex 403 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
7463, 73jaoi 846 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
75 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
76 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
77 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
78 vex 3401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
7975, 76, 77, 78preq12b 4610 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
8074, 79syl6ibr 244 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
8150, 80sylbi 209 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
8281com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))
8382expcom 404 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓:𝑉1-1-onto𝑊 → (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8483expd 406 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:𝑉1-1-onto𝑊 → ((𝑎𝑉𝑏𝑉) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
8584ad2antlr 717 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) → ((𝑎𝑉𝑏𝑉) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
8685imp 397 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8786adantr 474 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8887adantr 474 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8988imp31 410 . . . . . . . . . . . . . . . 16 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}) → {𝑥, 𝑦} = {𝑎, 𝑏})
9089eleq1d 2844 . . . . . . . . . . . . . . 15 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
9190biimpd 221 . . . . . . . . . . . . . 14 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}) → ({𝑥, 𝑦} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸))
9291impancom 445 . . . . . . . . . . . . 13 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9345, 92sylbid 232 . . . . . . . . . . . 12 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9493adantr 474 . . . . . . . . . . 11 ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9534, 94sylbid 232 . . . . . . . . . 10 ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9631, 95mpdan 677 . . . . . . . . 9 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9723, 96sylbird 252 . . . . . . . 8 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
9897impancom 445 . . . . . . 7 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}) → ({𝑥, 𝑦} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸))
9916, 98mpd 15 . . . . . 6 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}) → {𝑎, 𝑏} ∈ 𝐸)
10099ex 403 . . . . 5 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
101100rexlimdvva 3221 . . . 4 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → (∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
10211, 101mpd 15 . . 3 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → {𝑎, 𝑏} ∈ 𝐸)
1034, 102mpdan 677 . 2 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → {𝑎, 𝑏} ∈ 𝐸)
104103ex 403 1 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ({(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  {cpr 4400  ◡ccnv 5354   “ cima 5358   Fn wfn 6130  –1-1→wf1 6132  –1-1-onto→wf1o 6134  ‘cfv 6135  Vtxcvtx 26344  Edgcedg 26395  UPGraphcupgr 26428  USPGraphcuspgr 26497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-hash 13436  df-edg 26396  df-upgr 26430  df-uspgr 26499 This theorem is referenced by:  isomuspgr  42751
 Copyright terms: Public domain W3C validator