Theorem uhgrimedgi 49203

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 2763	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . . . . 6 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3		eqid 2763	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2763	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 49196	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
6		uhgrimedgi.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (Edg‘𝐺)
7	6	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ (Edg‘𝐺))
8	3	uhgrfun 29677	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	3	edgiedgb 29665	. . . . . . . . . . . . 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)⟶1-1onto(Vtx‘𝐻)) → 𝑘 ∈ dom (iEdg‘𝐺))
14		2fveq3 6916	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)))
15		fveq2 6911	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
16	15	imaeq2d 6065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
17	14, 16	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑘 → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
18	17	rspcv 3576	. . . . . . . . . . . . . . . . . . . 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)⟶1-1onto(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 29677	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
21	20	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝐻)) → Fun (iEdg‘𝐻))
22		f1of 6849	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → 𝑘 ∈ dom (iEdg‘𝐺))
25	23, 24	ffvelcdmd 7112	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → (𝑗‘𝑘) ∈ dom (iEdg‘𝐻))
26	4	iedgedg 29661	. . . . . . . . . . . . . . . . . . . . . . 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)⟶1-1onto(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ (Edg‘𝐻))
28		uhgrimedgi.d	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 = (Edg‘𝐻)
29	27, 28	eleqtrrdi 2875	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) ∧ 𝐹:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷)
30		eleq1 2852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) = ((iEdg‘𝐻)‘(𝑗‘𝑘)) → ((𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗‘𝑘)) ∈ 𝐷))
31	30	eqcoms 2771	. . . . . . . . . . . . . . . . . . . . 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)⟶1-1onto(Vtx‘𝐻)) ∧ 𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻)) → (𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻)) → (𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻)) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) ∧ (𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)
39		imaeq2 6061	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹 “ 𝐾) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
40	39	eleq1d 2849	. . . . . . . . . . . . . . . 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)⟶1-1onto(Vtx‘𝐻) ∧ (𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) ∧ (𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))))
46	45	rexlimdva 3167	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (∃𝑘 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑘) → (𝐹:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻)) → ((𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝐹 “ 𝐾) ∈ 𝐷))
50	49	exlimdv 1968	. . . . . 6 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐾 ∈ 𝐸) ∧ 𝐹:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝐻)) → (∃𝑗(𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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)⟶1-1onto(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)⟶1-1ontodom (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 2147  ∀wral 3080  ∃wrex 3090  dom cdm 5656   “ cima 5659  Fun wfun 6555  ⟶wf 6557  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  iEdgciedg 29608  Edgcedg 29658  UHGraphcuhgr 29667   GraphIso cgrim 49188

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880  df-edg 29659  df-uhgr 29669  df-grim 49191

This theorem is used by:  uhgrimedg  49204  upgrimwlklem2  49211  upgrimwlklem3  49212  upgrimtrlslem1  49217