isomuspgrlem1 - Mathbox for Alexander van der Vekens

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Theorem isomuspgrlem1 45279

Description: Lemma 1 for isomuspgr 45286. (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 483	. . . . 5 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐸–1-1-onto→𝐾)
2	1	ad2antlr 724	. . . 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 7157	. . . 4 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → (^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸)
4	2, 3	sylan 580	. . 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 27546	. . . . . . 7 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
6	5	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
7	6	ad4antr 729	. . . . 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 27507	. . . . 5 ⊢ ((𝐴 ∈ UPGraph ∧ (^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦})
11	7, 10	sylan 580	. . . 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 2826	. . . . . . . . . . 11 ⊢ ((^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → ((^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
13	12	biimpd 228	. . . . . . . . . 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 724	. . . . . . . 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 407	. . . . . . 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 734	. . . . . . . . . . 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 485	. . . . . . . . . . 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 485	. . . . . . . . . . . 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 753	. . . . . . . . . . 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 1127	. . . . . . . . . 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 7151	. . . . . . . . . 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 5965	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝑓 “ 𝑒) = (𝑓 “ {𝑥, 𝑦}))
25		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝑔‘𝑒) = (𝑔‘{𝑥, 𝑦}))
26	24, 25	eqeq12d 2754	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑥, 𝑦} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
27	26	rspccv 3558	. . . . . . . . . . . . . 14 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
28	27	adantl 482	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 729	. . . . . . . . . . 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 407	. . . . . . . . . 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 2742	. . . . . . . . . . . . 13 ⊢ ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
33	32	eqcoms 2746	. . . . . . . . . . . 12 ⊢ ((𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
34	33	adantl 482	. . . . . . . . . . 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 6717	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
36	35	ad6antlr 734	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
38	37	adantl 482	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
40	39	adantl 482	. . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 6852	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 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 6787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑥) ∈ V
47		fvex 6787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑦) ∈ V
48		fvex 6787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑎) ∈ V
49		fvex 6787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑏) ∈ V
50	46, 47, 48, 49	preq12b 4781	. . . . . . . . . . . . . . . . . . . . . . . . 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 6715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓:𝑉–1-1→𝑊)
52		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
53	52, 37	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
54		f1veqaeq 7130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → ((𝑓‘𝑥) = (𝑓‘𝑎) → 𝑥 = 𝑎))
55	51, 53, 54	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑓‘𝑥) = (𝑓‘𝑎) → 𝑥 = 𝑎))
56		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
57	56, 39	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
58		f1veqaeq 7130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓‘𝑦) = (𝑓‘𝑏) → 𝑦 = 𝑏))
59	51, 57, 58	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
65		f1veqaeq 7130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓‘𝑥) = (𝑓‘𝑏) → 𝑥 = 𝑏))
66	51, 64, 65	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
68		f1veqaeq 7130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → ((𝑓‘𝑦) = (𝑓‘𝑎) → 𝑦 = 𝑎))
69	51, 67, 68	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 871	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 854	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑥 ∈ V
76		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑦 ∈ V
77		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑎 ∈ V
78		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑏 ∈ V
79	75, 76, 77, 78	preq12b 4781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
80	74, 79	syl6ibr 251	. . . . . . . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . 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 452	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 239	. . . . . . . . . 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 684	. . . . . . . . 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 259	. . . . . . . 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 452	. . . . . . 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 413	. . . . 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 3223	. . . 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 684	. 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 413	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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {cpr 4563 ^◡ccnv 5588 “ cima 5592 Fn wfn 6428 –1-1→wf1 6430 –1-1-onto→wf1o 6432 ‘cfv 6433 Vtxcvtx 27366 Edgcedg 27417 UPGraphcupgr 27450 USPGraphcuspgr 27518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-hash 14045 df-edg 27418 df-upgr 27452 df-uspgr 27520

This theorem is referenced by: isomuspgr 45286

W3C validator