isomuspgrlem1 - Mathbox for Alexander van der Vekens

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Theorem isomuspgrlem1 44345

Description: Lemma 1 for isomuspgr 44352. (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 486	. . . . 5 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐸–1-1-onto→𝐾)
2	1	ad2antlr 726	. . . 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 7019	. . . 4 ⊢ ((𝑔:𝐸–1-1-onto→𝐾 ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → (^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸)
4	2, 3	sylan 583	. . 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 26969	. . . . . . 7 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
6	5	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
7	6	ad4antr 731	. . . . 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 26930	. . . . 5 ⊢ ((𝐴 ∈ UPGraph ∧ (^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦})
11	7, 10	sylan 583	. . . 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 2877	. . . . . . . . . . 11 ⊢ ((^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) = {𝑥, 𝑦} → ((^◡𝑔‘{(𝑓‘𝑎), (𝑓‘𝑏)}) ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
13	12	biimpd 232	. . . . . . . . . 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 726	. . . . . . . 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 410	. . . . . . 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 736	. . . . . . . . . . 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 488	. . . . . . . . . . 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 488	. . . . . . . . . . . 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 755	. . . . . . . . . . 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 1125	. . . . . . . . . 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 7014	. . . . . . . . . 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 5892	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝑓 “ 𝑒) = (𝑓 “ {𝑥, 𝑦}))
25		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝑔‘𝑒) = (𝑔‘{𝑥, 𝑦}))
26	24, 25	eqeq12d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑥, 𝑦} → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
27	26	rspccv 3568	. . . . . . . . . . . . . 14 ⊢ (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
28	27	adantl 485	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 731	. . . . . . . . . . 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 410	. . . . . . . . . 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 2802	. . . . . . . . . . . . 13 ⊢ ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
33	32	eqcoms 2806	. . . . . . . . . . . 12 ⊢ ((𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
34	33	adantl 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) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)}))
35		f1ofn 6591	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓 Fn 𝑉)
36	35	ad6antlr 736	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
38	37	adantl 485	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
40	39	adantl 485	. . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 6722	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑥) ∈ V
47		fvex 6658	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑦) ∈ V
48		fvex 6658	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑎) ∈ V
49		fvex 6658	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓‘𝑏) ∈ V
50	46, 47, 48, 49	preq12b 4741	. . . . . . . . . . . . . . . . . . . . . . . . 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 6589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓:𝑉–1-1→𝑊)
52		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
53	52, 37	anim12ci 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
54		f1veqaeq 6993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → ((𝑓‘𝑥) = (𝑓‘𝑎) → 𝑥 = 𝑎))
55	51, 53, 54	syl2anr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑓:𝑉–1-1-onto→𝑊) → ((𝑓‘𝑥) = (𝑓‘𝑎) → 𝑥 = 𝑎))
56		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
57	56, 39	anim12ci 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
58		f1veqaeq 6993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓‘𝑦) = (𝑓‘𝑏) → 𝑦 = 𝑏))
59	51, 57, 58	syl2anr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
65		f1veqaeq 6993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑓‘𝑥) = (𝑓‘𝑏) → 𝑥 = 𝑏))
66	51, 64, 65	syl2anr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
68		f1veqaeq 6993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝑉–1-1→𝑊 ∧ (𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) → ((𝑓‘𝑦) = (𝑓‘𝑎) → 𝑦 = 𝑎))
69	51, 67, 68	syl2anr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑥 ∈ V
76		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑦 ∈ V
77		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑎 ∈ V
78		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑏 ∈ V
79	75, 76, 77, 78	preq12b 4741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
80	74, 79	syl6ibr 255	. . . . . . . . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 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) → ({(𝑓‘𝑥), (𝑓‘𝑦)} = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
89	88	imp31 421	. . . . . . . . . . . . . . . 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 2874	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 455	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 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})) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓‘𝑎), (𝑓‘𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
95	34, 94	sylbid 243	. . . . . . . . . 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 686	. . . . . . . . 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 263	. . . . . . . 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 455	. . . . . . 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 416	. . . . 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 3253	. . . 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 686	. 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 416	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 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 {cpr 4527 ^◡ccnv 5518 “ cima 5522 Fn wfn 6319 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 Vtxcvtx 26789 Edgcedg 26840 UPGraphcupgr 26873 USPGraphcuspgr 26941

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-edg 26841 df-upgr 26875 df-uspgr 26943

This theorem is referenced by: isomuspgr 44352

W3C validator