Theorem uhgrimedgi 48014

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 2733	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . . . . 6 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3		eqid 2733	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2733	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 48007	. . . . 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 2825	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ (Edg‘𝐺))
8	3	uhgrfun 29046	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	3	edgiedgb 29034	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
11	7, 10	bitrid 283	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
12	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘)))
13		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → 𝑘 ∈ dom (iEdg‘𝐺))
14		2fveq3 6833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)))
15		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
16	15	imaeq2d 6013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
17	14, 16	eqeq12d 2749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑘 → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
18	17	rspcv 3569	. . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . 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 29046	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
21	20	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)) → Fun (iEdg‘𝐻))
22		f1of 6768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
23	22	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 7024	. . . . . . . . . . . . . . . . . . . . . . 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 29030	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun (iEdg‘𝐻) ∧ (𝑗‘𝑘) ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
27	21, 25, 26	syl2an2r 685	. . . . . . . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
31	30	eqcoms 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
32	29, 31	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 73	. . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . . 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 6009	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝐾) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
40	39	eleq1d 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
41	40	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑘)) → ((𝐹 “ 𝐾) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
42	41	3ad2ant1 1133	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 1119	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 3134	. . . . . . . . . 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 240	. . . . . . . . 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 406	. . . . . . . 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 406	. . . . . . 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 1934	. . . . . 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 453	. . . . 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 34	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹 “ 𝐾) ∈ 𝐷))
53	52	ex 412	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐾 ∈ 𝐸 → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹 “ 𝐾) ∈ 𝐷)))
54	53	impcomd 411	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → ((𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸) → (𝐹 “ 𝐾) ∈ 𝐷))
55	54	imp 406	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  dom cdm 5619   “ cima 5622  Fun wfun 6480  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7352  Vtxcvtx 28976  iEdgciedg 28977  Edgcedg 29027  UHGraphcuhgr 29036   GraphIso cgrim 47999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-edg 29028  df-uhgr 29038  df-grim 48002

This theorem is referenced by:  uhgrimedg  48015  upgrimwlklem2  48022  upgrimwlklem3  48023  upgrimtrlslem1  48028