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Theorem isomuspgrlem1 45258
Description: Lemma 1 for isomuspgr 45265. (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 483 . . . . 5 ((𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)) → 𝑔:𝐸1-1-onto𝐾)
21ad2antlr 724 . . . 4 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → 𝑔:𝐸1-1-onto𝐾)
3 f1ocnvdm 7154 . . . 4 ((𝑔:𝐸1-1-onto𝐾 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸)
42, 3sylan 580 . . 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 27557 . . . . . . 7 (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
65adantr 481 . . . . . 6 ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
76ad4antr 729 . . . . 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 27518 . . . . 5 ((𝐴 ∈ UPGraph ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦})
117, 10sylan 580 . . . 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 2828 . . . . . . . . . . 11 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
1312biimpd 228 . . . . . . . . . 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 724 . . . . . . . 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 407 . . . . . . 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 734 . . . . . . . . . . 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 485 . . . . . . . . . . 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 485 . . . . . . . . . . . 12 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾)
2019ad5ant12 753 . . . . . . . . . . 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 1127 . . . . . . . . . 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 7148 . . . . . . . . . 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 5964 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑓𝑒) = (𝑓 “ {𝑥, 𝑦}))
25 fveq2 6771 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑔𝑒) = (𝑔‘{𝑥, 𝑦}))
2624, 25eqeq12d 2756 . . . . . . . . . . . . . . 15 (𝑒 = {𝑥, 𝑦} → ((𝑓𝑒) = (𝑔𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2726rspccv 3558 . . . . . . . . . . . . . 14 (∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2827adantl 482 . . . . . . . . . . . . 13 ((𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2928adantl 482 . . . . . . . . . . . 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 729 . . . . . . . . . . 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 407 . . . . . . . . . 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 2744 . . . . . . . . . . . . 13 ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3332eqcoms 2748 . . . . . . . . . . . 12 ((𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3433adantl 482 . . . . . . . . . . 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 6715 . . . . . . . . . . . . . . . . . 18 (𝑓:𝑉1-1-onto𝑊𝑓 Fn 𝑉)
3635ad6antlr 734 . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑦𝑉) → 𝑥𝑉)
3837adantl 482 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑦𝑉) → 𝑦𝑉)
4039adantl 482 . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . 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 6784 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑥) ∈ V
47 fvex 6784 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑦) ∈ V
48 fvex 6784 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑎) ∈ V
49 fvex 6784 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑏) ∈ V
5046, 47, 48, 49preq12b 4787 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} ↔ (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))))
51 f1of1 6713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:𝑉1-1-onto𝑊𝑓:𝑉1-1𝑊)
52 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑎𝑉)
5352, 37anim12ci 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑎𝑉))
54 f1veqaeq 7127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑥𝑉𝑎𝑉)) → ((𝑓𝑥) = (𝑓𝑎) → 𝑥 = 𝑎))
5551, 53, 54syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑥) = (𝑓𝑎) → 𝑥 = 𝑎))
56 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑏𝑉)
5756, 39anim12ci 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑦𝑉𝑏𝑉))
58 f1veqaeq 7127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑦𝑉𝑏𝑉)) → ((𝑓𝑦) = (𝑓𝑏) → 𝑦 = 𝑏))
5951, 57, 58syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑦) = (𝑓𝑏) → 𝑦 = 𝑏))
6055, 59anim12d 609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) → (𝑥 = 𝑎𝑦 = 𝑏)))
6160impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → (𝑥 = 𝑎𝑦 = 𝑏))
6261orcd 870 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
6362ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
6456, 37anim12ci 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑏𝑉))
65 f1veqaeq 7127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑥𝑉𝑏𝑉)) → ((𝑓𝑥) = (𝑓𝑏) → 𝑥 = 𝑏))
6651, 64, 65syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑥) = (𝑓𝑏) → 𝑥 = 𝑏))
6752, 39anim12ci 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑦𝑉𝑎𝑉))
68 f1veqaeq 7127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑦𝑉𝑎𝑉)) → ((𝑓𝑦) = (𝑓𝑎) → 𝑦 = 𝑎))
6951, 67, 68syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑦) = (𝑓𝑎) → 𝑦 = 𝑎))
7066, 69anim12d 609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → (((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) → (𝑥 = 𝑏𝑦 = 𝑎)))
7170impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → (𝑥 = 𝑏𝑦 = 𝑎))
7271olcd 871 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
7372ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
7463, 73jaoi 854 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
75 vex 3435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
76 vex 3435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
77 vex 3435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
78 vex 3435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
7975, 76, 77, 78preq12b 4787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
8074, 79syl6ibr 251 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
8150, 80sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓:𝑉1-1-onto𝑊 → (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8483expd 416 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:𝑉1-1-onto𝑊 → ((𝑎𝑉𝑏𝑉) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
8584ad2antlr 724 . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 239 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 239 . . . . . . . . . 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 684 . . . . . . . . 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 259 . . . . . . . 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 452 . . . . . . 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 413 . . . . 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 3225 . . . 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 684 . 2 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → {𝑎, 𝑏} ∈ 𝐸)
104103ex 413 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 205  wa 396  wo 844  w3a 1086   = wceq 1542  wcel 2110  wral 3066  wrex 3067  {cpr 4569  ccnv 5589  cima 5593   Fn wfn 6427  1-1wf1 6429  1-1-ontowf1o 6431  cfv 6432  Vtxcvtx 27377  Edgcedg 27428  UPGraphcupgr 27461  USPGraphcuspgr 27529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-om 7708  df-1st 7825  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-1o 8289  df-2o 8290  df-oadd 8293  df-er 8490  df-en 8726  df-dom 8727  df-sdom 8728  df-fin 8729  df-dju 9670  df-card 9708  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-nn 11985  df-2 12047  df-n0 12245  df-xnn0 12317  df-z 12331  df-uz 12594  df-fz 13251  df-hash 14056  df-edg 27429  df-upgr 27463  df-uspgr 27531
This theorem is referenced by:  isomuspgr  45265
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