Theorem isomuspgrlem1 43999

Description: Lemma 1 for isomuspgr 44006. (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:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 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.

Step	Hyp	Ref	Expression
1		simpl 485	. . . . 5 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐸–1-1-onto→𝐾)
2	1	ad2antlr 725	. . . 4 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑔:𝐸–1-1-onto→𝐾)
3		f1ocnvdm 7044	. . . 4 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸)
4	2, 3	sylan 582	. . 3 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸)
5		uspgrupgr 26964	. . . . . . 7 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
6	5	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
7	6	ad4antr 730	. . . . 5 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → 𝐴 ∈ UPGraph)
8		isomushgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐴)
9		isomushgr.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐴)
10	8, 9	upgredg 26925	. . . . 5 ⊢ ((𝐴 ∈ UPGraph ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦})
11	7, 10	sylan 582	. . . 4 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦})
12		eleq1 2903	. . . . . . . . . . 11 ⊢ ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
13	12	biimpd 231	. . . . . . . . . 10 ⊢ ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸 → {𝑥, 𝑦} ∈ 𝐸))
14	13	com12 32	. . . . . . . . 9 ⊢ ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸 → ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → {𝑥, 𝑦} ∈ 𝐸))
15	14	ad2antlr 725	. . . . . . . 8 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → {𝑥, 𝑦} ∈ 𝐸))
16	15	imp 409	. . . . . . 7 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦}) → {𝑥, 𝑦} ∈ 𝐸)
17	1	ad6antlr 735	. . . . . . . . . . 11 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑔:𝐸–1-1-onto→𝐾)
18		simpr 487	. . . . . . . . . . 11 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → {𝑥, 𝑦} ∈ 𝐸)
19		simpr 487	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)
20	19	ad5ant12 754	. . . . . . . . . . 11 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)
21	17, 18, 20	3jca 1124	. . . . . . . . . 10 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑔:𝐸–1-1-onto→𝐾 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
22		f1ocnvfvb 7039	. . . . . . . . . 10 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦}))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦}))
24		imaeq2 5928	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝑓 “ 𝑒) = (𝑓 “ {𝑥, 𝑦}))
25		fveq2 6673	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝑔‘𝑒) = (𝑔‘{𝑥, 𝑦}))
26	24, 25	eqeq12d 2840	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑥, 𝑦} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
27	26	rspccv 3623	. . . . . . . . . . . . . 14 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
28	27	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
29	28	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
30	29	ad4antr 730	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
31	30	imp 409	. . . . . . . . . 10 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}))
32		eqeq1 2828	. . . . . . . . . . . . 13 ⊢ ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
33	32	eqcoms 2832	. . . . . . . . . . . 12 ⊢ ((𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
34	33	adantl 484	. . . . . . . . . . 11 ⊢ ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
35		f1ofn 6619	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
36	35	ad6antlr 735	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑓 Fn 𝑉)
37		simpl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
38	37	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
39		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
40	39	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
41	36, 38, 40	3jca 1124	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑓 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
42	41	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
43		fnimapr 6750	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑥), (𝑓‘𝑦)})
44	42, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑥), (𝑓‘𝑦)})
45	44	eqeq1d 2826	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ {(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)}))
46		fvex 6686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑥) ∈ V
47		fvex 6686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑦) ∈ V
48		fvex 6686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑎) ∈ V
49		fvex 6686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑏) ∈ V
50	46, 47, 48, 49	preq12b 4784	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) ∨ ((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎))))
51		f1of1 6617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓:𝑉–1-1→𝑊)
52		simpl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
53	52, 37	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
54		f1veqaeq 7018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → ((𝑓‘𝑥) = (𝑓‘𝑎) → 𝑥 = 𝑎))
55	51, 53, 54	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑓‘𝑥) = (𝑓‘𝑎) → 𝑥 = 𝑎))
56		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
57	56, 39	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
58		f1veqaeq 7018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓‘𝑦) = (𝑓‘𝑏) → 𝑦 = 𝑏))
59	51, 57, 58	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑓‘𝑦) = (𝑓‘𝑏) → 𝑦 = 𝑏))
60	55, 59	anim12d 610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) → (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)))
61	60	impcom 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) ∧ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊)) → (𝑥 = 𝑎 ∧ 𝑦 = 𝑏))
62	61	orcd 869	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) ∧ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊)) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
63	62	ex 415	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) → ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎))))
64	56, 37	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
65		f1veqaeq 7018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓‘𝑥) = (𝑓‘𝑏) → 𝑥 = 𝑏))
66	51, 64, 65	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑓‘𝑥) = (𝑓‘𝑏) → 𝑥 = 𝑏))
67	52, 39	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
68		f1veqaeq 7018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → ((𝑓‘𝑦) = (𝑓‘𝑎) → 𝑦 = 𝑎))
69	51, 67, 68	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑓‘𝑦) = (𝑓‘𝑎) → 𝑦 = 𝑎))
70	66, 69	anim12d 610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎)) → (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
71	70	impcom 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎)) ∧ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊)) → (𝑥 = 𝑏 ∧ 𝑦 = 𝑎))
72	71	olcd 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎)) ∧ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊)) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
73	72	ex 415	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎)) → ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎))))
74	63, 73	jaoi 853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) ∨ ((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎))) → ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎))))
75		vex 3500	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑥 ∈ V
76		vex 3500	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑦 ∈ V
77		vex 3500	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑎 ∈ V
78		vex 3500	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑏 ∈ V
79	75, 76, 77, 78	preq12b 4784	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
80	74, 79	syl6ibr 254	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑓‘𝑥) = (𝑓‘𝑎) ∧ (𝑓‘𝑦) = (𝑓‘𝑏)) ∨ ((𝑓‘𝑥) = (𝑓‘𝑏) ∧ (𝑓‘𝑦) = (𝑓‘𝑎))) → ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
81	50, 80	sylbi 219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
82	81	com12 32	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))
83	82	expcom 416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
84	83	expd 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
85	84	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
86	85	imp 409	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
87	86	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
88	87	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
89	88	imp31 420	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)}) → {𝑥, 𝑦} = {𝑎, 𝑏})
90	89	eleq1d 2900	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)}) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
91	90	biimpd 231	. . . . . . . . . . . . . 14 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)}) → ({𝑥, 𝑦} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸))
92	91	impancom 454	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
93	45, 92	sylbid 242	. . . . . . . . . . . 12 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
94	93	adantr 483	. . . . . . . . . . 11 ⊢ ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
95	34, 94	sylbid 242	. . . . . . . . . 10 ⊢ ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
96	31, 95	mpdan 685	. . . . . . . . 9 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
97	23, 96	sylbird 262	. . . . . . . 8 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
98	97	impancom 454	. . . . . . 7 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦}) → ({𝑥, 𝑦} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸))
99	16, 98	mpd 15	. . . . . 6 ⊢ (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦}) → {𝑎, 𝑏} ∈ 𝐸)
100	99	ex 415	. . . . 5 ⊢ ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
101	100	rexlimdvva 3297	. . . 4 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
102	11, 101	mpd 15	. . 3 ⊢ (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) ∧ (◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → {𝑎, 𝑏} ∈ 𝐸)
103	4, 102	mpdan 685	. 2 ⊢ ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → {𝑎, 𝑏} ∈ 𝐸)
104	103	ex 415	1 ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  {cpr 4572  ^◡ccnv 5557   “ cima 5561   Fn wfn 6353  –1-1→wf1 6355  –1-1-onto→wf1o 6357  ‘cfv 6358  Vtxcvtx 26784  Edgcedg 26835  UPGraphcupgr 26868  USPGraphcuspgr 26936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-fz 12896  df-hash 13694  df-edg 26836  df-upgr 26870  df-uspgr 26938

This theorem is referenced by:  isomuspgr  44006