Theorem uhgrimedgi 48510

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 2765	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2765	. . . . . 6 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3		eqid 2765	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2765	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 48503	. . . . 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 2857	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ (Edg‘𝐺))
8	3	uhgrfun 29325	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	3	edgiedgb 29313	. . . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
13		simplr 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → 𝑘 ∈ dom (iEdg‘𝐺))
14		2fveq3 6876	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)))
15		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
16	15	imaeq2d 6053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
17	14, 16	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . 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 29325	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
21	20	ad3antlr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → Fun (iEdg‘𝐻))
22		f1of 6810	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
23	22	adantl 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)) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
24	13	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 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 7070	. . . . . . . . . . . . . . . . . . . . . . 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 29309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun (iEdg‘𝐻) ∧ (𝑗‘𝑘) ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
27	21, 25, 26	syl2an2r 697	. . . . . . . . . . . . . . . . . . . . . 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 2876	. . . . . . . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
31	30	eqcoms 2773	. . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . . 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 6049	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝐾) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
40	39	eleq1d 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
41	40	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
42	41	3ad2ant1 1149	. . . . . . . . . . . . . 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 1135	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 3166	. . . . . . . . . 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 411	. . . . . . . 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 411	. . . . . . 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 1956	. . . . . 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 458	. . . . 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 417	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹 “ 𝐾) ∈ 𝐷)))
54	53	impcomd 416	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸) → (𝐹 “ 𝐾) ∈ 𝐷))
55	54	imp 411	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  dom cdm 5652   “ cima 5655  Fun wfun 6519  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  Vtxcvtx 29255  iEdgciedg 29256  Edgcedg 29306  UHGraphcuhgr 29315   GraphIso cgrim 48495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-edg 29307  df-uhgr 29317  df-grim 48498

This theorem is referenced by:  uhgrimedg  48511  upgrimwlklem2  48518  upgrimwlklem3  48519  upgrimtrlslem1  48524