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Theorem isomuspgrlem1 46009
Description: Lemma 1 for isomuspgr 46016. (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 725 . . . 4 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → 𝑔:𝐸1-1-onto𝐾)
3 f1ocnvdm 7231 . . . 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 28127 . . . . . . 7 (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
65adantr 481 . . . . . 6 ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
76ad4antr 730 . . . . 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 28088 . . . . 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 2825 . . . . . . . . . . 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 725 . . . . . . . 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 735 . . . . . . . . . . 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 754 . . . . . . . . . . 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 1128 . . . . . . . . . 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 7225 . . . . . . . . . 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 6009 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑓𝑒) = (𝑓 “ {𝑥, 𝑦}))
25 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑔𝑒) = (𝑔‘{𝑥, 𝑦}))
2624, 25eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑒 = {𝑥, 𝑦} → ((𝑓𝑒) = (𝑔𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2726rspccv 3578 . . . . . . . . . . . . . 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 730 . . . . . . . . . . 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 2740 . . . . . . . . . . . . 13 ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3332eqcoms 2744 . . . . . . . . . . . 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 6785 . . . . . . . . . . . . . . . . . 18 (𝑓:𝑉1-1-onto𝑊𝑓 Fn 𝑉)
3635ad6antlr 735 . . . . . . . . . . . . . . . . 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 1128 . . . . . . . . . . . . . . . 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 6925 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑥) ∈ V
47 fvex 6855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑦) ∈ V
48 fvex 6855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑎) ∈ V
49 fvex 6855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑏) ∈ V
5046, 47, 48, 49preq12b 4808 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} ↔ (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))))
51 f1of1 6783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:𝑉1-1-onto𝑊𝑓:𝑉1-1𝑊)
52 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑎𝑉)
5352, 37anim12ci 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑎𝑉))
54 f1veqaeq 7204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 871 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 855 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
75 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
76 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
77 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
78 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
7975, 76, 77, 78preq12b 4808 . . . . . . . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . 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 685 . . . . . . . . 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 3205 . . . 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 685 . 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 845  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {cpr 4588  ccnv 5632  cima 5636   Fn wfn 6491  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  Vtxcvtx 27947  Edgcedg 27998  UPGraphcupgr 28031  USPGraphcuspgr 28099
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-upgr 28033  df-uspgr 28101
This theorem is referenced by:  isomuspgr  46016
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