Theorem uhgrimedgi 48696

Description: An isomorphism between graphs preserves edges, i.e. if there is an edge in one graph connecting vertices then there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)

Hypotheses

Ref	Expression
uhgrimedgi.e	⊢ 𝐸 = (Edg‘𝐺)
uhgrimedgi.d	⊢ 𝐷 = (Edg‘𝐻)

Assertion

Ref	Expression
uhgrimedgi	⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)

Proof of Theorem uhgrimedgi

Dummy variables 𝑗 𝑘 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2766	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2766	. . . . . 6 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3		eqid 2766	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2766	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 48689	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
6		uhgrimedgi.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (Edg‘𝐺)
7	6	eleq2i 2858	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ (Edg‘𝐺))
8	3	uhgrfun 29453	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	3	edgiedgb 29441	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
10	8, 9	syl 18	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
11	7, 10	bitrid 286	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
12	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
13		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → 𝑘 ∈ dom (iEdg‘𝐺))
14		2fveq3 6893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)))
15		fveq2 6888	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
16	15	imaeq2d 6067	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
17	14, 16	eqeq12d 2782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑘 → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
18	17	rspcv 3580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
19	13, 18	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
20	4	uhgrfun 29453	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
21	20	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → Fun (iEdg‘𝐻))
22		f1of 6827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
23	22	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
24	13	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → 𝑘 ∈ dom (iEdg‘𝐺))
25	23, 24	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → (𝑗‘𝑘) ∈ dom (iEdg‘𝐻))
26	4	iedgedg 29437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun (iEdg‘𝐻) ∧ (𝑗‘𝑘) ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
27	21, 25, 26	syl2an2r 698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
28		uhgrimedgi.d	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 = (Edg‘𝐻)
29	27, 28	eleqtrrdi 2877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷)
30		eleq1 2854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
31	30	eqcoms 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
32	29, 31	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
33	32	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → (((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
34	19, 33	syl5d 74	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
35	34	impd 416	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
36	35	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
37	36	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
38	37	3imp 1128	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)
39		imaeq2 6063	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝐾) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
40	39	eleq1d 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
41	40	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
42	41	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
43	38, 42	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝐹 “ 𝐾) ∈ 𝐷)
44	43	3exp 1137	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷)))
45	44	ex 418	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))))
46	45	rexlimdva 3169	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))))
47	12, 46	sylbid 243	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))))
48	47	imp 412	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷)))
49	48	imp 412	. . . . . . 7 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))
50	49	exlimdv 1966	. . . . . 6 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))
51	50	expimpd 459	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) → ((𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝐹 “ 𝐾) ∈ 𝐷))
52	5, 51	syl5 35	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹 “ 𝐾) ∈ 𝐷))
53	52	ex 418	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹 “ 𝐾) ∈ 𝐷)))
54	53	impcomd 417	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸) → (𝐹 “ 𝐾) ∈ 𝐷))
55	54	imp 412	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  dom cdm 5666   “ cima 5669  Fun wfun 6537  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Vtxcvtx 29383  iEdgciedg 29384  Edgcedg 29434  UHGraphcuhgr 29443   GraphIso cgrim 48681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-edg 29435  df-uhgr 29445  df-grim 48684

This theorem is used by:  uhgrimedg  48697  upgrimwlklem2  48704  upgrimwlklem3  48705  upgrimtrlslem1  48710