Theorem uhgrimedgi 49081

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 2760	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2760	. . . . . 6 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3		eqid 2760	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2760	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 49074	. . . . 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 2852	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ (Edg‘𝐺))
8	3	uhgrfun 29649	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	3	edgiedgb 29637	. . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)))
15		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
16	15	imaeq2d 6051	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
17	14, 16	eqeq12d 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑘 → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
18	17	rspcv 3572	. . . . . . . . . . . . . . . . . . . 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 29649	. . . . . . . . . . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . . . . . . . . . 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 29633	. . . . . . . . . . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
31	30	eqcoms 2768	. . . . . . . . . . . . . . . . . . . . 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 6047	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝐾) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
40	39	eleq1d 2845	. . . . . . . . . . . . . . . 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 3163	. . . . . . . . . 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 1965	. . . . . 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 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  dom cdm 5650   “ cima 5653  Fun wfun 6532  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29579  iEdgciedg 29580  Edgcedg 29630  UHGraphcuhgr 29639   GraphIso cgrim 49066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8852  df-edg 29631  df-uhgr 29641  df-grim 49069

This theorem is used by:  uhgrimedg  49082  upgrimwlklem2  49089  upgrimwlklem3  49090  upgrimtrlslem1  49095