isuspgrim0 - Mathbox for Alexander van der Vekens

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Theorem isuspgrim0 47830

Description: An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)

**Hypotheses**
Ref	Expression
isusgrim.v	⊢ 𝑉 = (Vtx‘𝐺)
isusgrim.w	⊢ 𝑊 = (Vtx‘𝐻)
isusgrim.e	⊢ 𝐸 = (Edg‘𝐺)
isusgrim.d	⊢ 𝐷 = (Edg‘𝐻)

**Assertion**
Ref	Expression
isuspgrim0	⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))

Distinct variable groups: 𝐷,𝑒 𝑒,𝐸 𝑒,𝐹 𝑒,𝐺 𝑒,𝐻 𝑒,𝑉 𝑒,𝑊 𝑒,𝑋

Proof of Theorem isuspgrim0 Dummy variables 𝑑 𝑖 𝑥 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isusgrim.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		isusgrim.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐻)
3		eqid 2734	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2734	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	isgrim 47826	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))))
6		isusgrim.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
7	6	eleq2i 2825	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
8		uspgruhgr 29129	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
9	3	uhgredgiedgb 29071	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
10	8, 9	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USPGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
11	7, 10	bitrid 283	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USPGraph → (𝑒 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
12	11	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝑒 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
13	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑒 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
14	13	biimpa 476	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒 ∈ 𝐸) → ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘))
15		2fveq3 6891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)))
16		fveq2 6886	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
17	16	imaeq2d 6058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
18	15, 17	eqeq12d 2750	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑘 → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
19	18	rspcv 3601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
20	19	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
21		uspgruhgr 29129	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UHGraph)
22	4	uhgrfun 29011	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐻 ∈ USPGraph → Fun (iEdg‘𝐻))
24	23	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → Fun (iEdg‘𝐻))
25	24	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → Fun (iEdg‘𝐻))
26		f1of 6828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
27	26	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
28	27	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝑗‘𝑘) ∈ dom (iEdg‘𝐻))
29	4	iedgedg 28995	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun (iEdg‘𝐻) ∧ (𝑗‘𝑘) ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
30	25, 28, 29	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
31		isusgrim.d	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 = (Edg‘𝐻)
32	31	eleq2i 2825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
33	30, 32	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷)
34		eleq1 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → (((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
35	33, 34	syl5ibcom 245	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
36	20, 35	syld 47	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
37	36	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → (𝑘 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
38	37	com23 86	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝑘 ∈ dom (iEdg‘𝐺) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
39	38	impr 454	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑘 ∈ dom (iEdg‘𝐺) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
40	39	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒 ∈ 𝐸) → (𝑘 ∈ dom (iEdg‘𝐺) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
41	40	imp 406	. . . . . . . . . . . 12 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒 ∈ 𝐸) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)
42		imaeq2 6054	. . . . . . . . . . . . 13 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝑒) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
43	42	eleq1d 2818	. . . . . . . . . . . 12 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑘) → ((𝐹 “ 𝑒) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
44	41, 43	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒 ∈ 𝐸) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝑒 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝑒) ∈ 𝐷))
45	44	rexlimdva 3142	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒 ∈ 𝐸) → (∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝑒) ∈ 𝐷))
46	14, 45	mpd 15	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒 ∈ 𝐸) → (𝐹 “ 𝑒) ∈ 𝐷)
47	46	ralrimiva 3133	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) ∈ 𝐷)
48	31	eleq2i 2825	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐷 ↔ 𝑑 ∈ (Edg‘𝐻))
49	4	uhgredgiedgb 29071	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UHGraph → (𝑑 ∈ (Edg‘𝐻) ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
50	21, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ USPGraph → (𝑑 ∈ (Edg‘𝐻) ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
51	48, 50	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ USPGraph → (𝑑 ∈ 𝐷 ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
52	51	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝑑 ∈ 𝐷 ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
53	52	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑑 ∈ 𝐷 ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
54		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
55		f1ocnvdm 7287	. . . . . . . . . . . . 13 ⊢ ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺))
56	54, 55	sylan 580	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺))
57		2fveq3 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (^◡𝑗‘𝑘) → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))))
58		fveq2 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (^◡𝑗‘𝑘) → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))
59	58	imaeq2d 6058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (^◡𝑗‘𝑘) → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))))
60	57, 59	eqeq12d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (^◡𝑗‘𝑘) → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
61	60	rspccv 3602	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
62	61	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → ((^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
63	62	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ((^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
64	63	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ((^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
65		f1ocnvfv2 7279	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑗‘(^◡𝑗‘𝑘)) = 𝑘)
66	54, 65	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑗‘(^◡𝑗‘𝑘)) = 𝑘)
67	66	fveqeq2d 6894	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) ↔ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
68		eqeq2 2746	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) ↔ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
69	68	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) ↔ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))))
70		simpll1 1212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → 𝐺 ∈ USPGraph)
71	6, 3	uspgriedgedg 29121	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USPGraph ∧ (^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺)) → ∃!𝑒 ∈ 𝐸 𝑒 = ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))
72	70, 56, 71	syl2an2r 685	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ∃!𝑒 ∈ 𝐸 𝑒 = ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))
73		eqcom 2741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) = 𝑒 ↔ 𝑒 = ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))
74	73	reubii 3372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑒 ∈ 𝐸 ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) = 𝑒 ↔ ∃!𝑒 ∈ 𝐸 𝑒 = ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))
75	72, 74	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ∃!𝑒 ∈ 𝐸 ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) = 𝑒)
76		f1of1 6827	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊)
77	76	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → 𝐹:𝑉–1-1→𝑊)
78		uspgrupgr 29123	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
79	78	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → 𝐺 ∈ UPGraph)
80	79	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → 𝐺 ∈ UPGraph)
81	80, 56	jca 511	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝐺 ∈ UPGraph ∧ (^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺)))
82	81	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → (𝐺 ∈ UPGraph ∧ (^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺)))
83	1, 3	upgrss 29033	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UPGraph ∧ (^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) ⊆ 𝑉)
84	82, 83	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) ⊆ 𝑉)
85	7	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
86		edgupgr 29079	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
87	80, 85, 86	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
88	87	simp1d 1142	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
89	88	elpwid 4589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ (Vtx‘𝐺))
90	89, 1	sseqtrrdi 4005	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
91		f1imaeq 7267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ (((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) ⊆ 𝑉 ∧ 𝑒 ⊆ 𝑉)) → ((𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒) ↔ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) = 𝑒))
92	77, 84, 90, 91	syl12anc 836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒 ∈ 𝐸) → ((𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒) ↔ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) = 𝑒))
93	92	reubidva 3379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (∃!𝑒 ∈ 𝐸 (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)) = 𝑒))
94	75, 93	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ∃!𝑒 ∈ 𝐸 (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒))
95	94	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) ∧ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) → ∃!𝑒 ∈ 𝐸 (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒))
96		eqeq1 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) → (𝑑 = (𝐹 “ 𝑒) ↔ (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒)))
97	96	reubidv 3381	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) → (∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒)))
98	97	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) ∧ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) → (∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) = (𝐹 “ 𝑒)))
99	95, 98	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) ∧ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))
100	99	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) → (𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
101	69, 100	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘)))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
102	101	ex 412	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
103	67, 102	sylbid 240	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘(𝑗‘(^◡𝑗‘𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(^◡𝑗‘𝑘))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
104	64, 103	syld 47	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ((^◡𝑗‘𝑘) ∈ dom (iEdg‘𝐺) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
105	56, 104	mpd 15	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
106	105	rexlimdva 3142	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
107	53, 106	sylbid 240	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
108	107	ralrimiv 3132	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ∀𝑑 ∈ 𝐷 ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))
109		imaeq2 6054	. . . . . . . . . 10 ⊢ (𝑥 = 𝑒 → (𝐹 “ 𝑥) = (𝐹 “ 𝑒))
110	109	cbvmptv 5235	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) = (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))
111	110	f1ompt 7111	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷 ↔ (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) ∈ 𝐷 ∧ ∀𝑑 ∈ 𝐷 ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
112	47, 108, 111	sylanbrc 583	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷)
113	112	ex 412	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷))
114	113	exlimdv 1932	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷))
115		fvex 6899	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) ∈ V
116	115	dmex 7913	. . . . . . . . 9 ⊢ dom (iEdg‘𝐺) ∈ V
117	116	mptex 7225	. . . . . . . 8 ⊢ (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) ∈ V
118	117	a1i 11	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷) → (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) ∈ V)
119		eqid 2734	. . . . . . . 8 ⊢ (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) = (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒))))
120	1, 2, 6, 31, 3, 4, 110, 119	isuspgrim0lem 47829	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷) → ((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
121		f1oeq1 6816	. . . . . . . 8 ⊢ (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
122		fveq1 6885	. . . . . . . . . 10 ⊢ (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (𝑗‘𝑖) = ((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖))
123	122	fveqeq2d 6894	. . . . . . . . 9 ⊢ (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
124	123	ralbidv 3165	. . . . . . . 8 ⊢ (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
125	121, 124	anbi12d 632	. . . . . . 7 ⊢ (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) ↔ ((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒)))):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ (^◡(iEdg‘𝐻)‘((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
126	118, 120, 125	spcedv 3581	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷) → ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
127	126	ex 412	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷 → ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
128	114, 127	impbid 212	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) ↔ (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷))
129		f1oeq1 6816	. . . . 5 ⊢ ((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) = (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) → ((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷 ↔ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷))
130	110, 129	mp1i 13	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)):𝐸–1-1-onto→𝐷 ↔ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷))
131	128, 130	bitrd 279	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) ↔ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷))
132	131	pm5.32da 579	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → ((𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
133	5, 132	bitrd 279	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 ∃!wreu 3361 Vcvv 3463 ⊆ wss 3931 ∅c0 4313 𝒫 cpw 4580 class class class wbr 5123 ↦ cmpt 5205 ^◡ccnv 5664 dom cdm 5665 “ cima 5668 Fun wfun 6535 ⟶wf 6537 –1-1→wf1 6538 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7413 ≤ cle 11278 2c2 12303 ♯chash 14351 Vtxcvtx 28941 iEdgciedg 28942 Edgcedg 28992 UHGraphcuhgr 29001 UPGraphcupgr 29025 USPGraphcuspgr 29093 GraphIso cgrim 47819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7416 df-oprab 7417 df-mpo 7418 df-map 8850 df-edg 28993 df-uhgr 29003 df-upgr 29027 df-uspgr 29095 df-grim 47822

This theorem is referenced by: uspgrimprop 47831 isuspgrim 47833

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