Theorem isomgrtrlem 46967

Description: Lemma for isomgrtr 46968. (Contributed by AV, 5-Dec-2022.)

Assertion

Ref	Expression
isomgrtrlem	⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))

Distinct variable groups:   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗,𝑘   𝐶,𝑗,𝑘   𝑗,𝑋   𝑓,𝑖,𝑗   𝑔,𝑖,𝑗,𝑘   𝑣,𝑗,𝑘   𝑤,𝑗,𝑘

Allowed substitution hints:   𝐴(𝑤,𝑣,𝑓,𝑔,𝑘)   𝐵(𝑤,𝑣,𝑓,𝑔)   𝐶(𝑤,𝑣,𝑓,𝑔,𝑖)   𝑋(𝑤,𝑣,𝑓,𝑔,𝑖,𝑘)

Proof of Theorem isomgrtrlem

Step	Hyp	Ref	Expression
1		imaco 6250	. . . 4 ⊢ ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = (𝑣 “ (𝑓 “ ((iEdg‘𝐴)‘𝑗)))
2	1	a1i 11	. . 3 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = (𝑣 “ (𝑓 “ ((iEdg‘𝐴)‘𝑗))))
3		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((iEdg‘𝐴)‘𝑖) = ((iEdg‘𝐴)‘𝑗))
4	3	imaeq2d 6059	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (𝑓 “ ((iEdg‘𝐴)‘𝑗)))
5		2fveq3 6896	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((iEdg‘𝐵)‘(𝑔‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑗)))
6	4, 5	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ (𝑓 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐵)‘(𝑔‘𝑗))))
7	6	rspccv 3609	. . . . . . . 8 ⊢ (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑓 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐵)‘(𝑔‘𝑗))))
8	7	adantl 481	. . . . . . 7 ⊢ ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑓 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐵)‘(𝑔‘𝑗))))
9	8	ad2antlr 724	. . . . . 6 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑓 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐵)‘(𝑔‘𝑗))))
10	9	imp 406	. . . . 5 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑓 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐵)‘(𝑔‘𝑗)))
11	10	imaeq2d 6059	. . . 4 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑣 “ (𝑓 “ ((iEdg‘𝐴)‘𝑗))) = (𝑣 “ ((iEdg‘𝐵)‘(𝑔‘𝑗))))
12		simplrr 775	. . . . 5 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))
13		f1of 6833	. . . . . . . . 9 ⊢ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → 𝑔:dom (iEdg‘𝐴)⟶dom (iEdg‘𝐵))
14		ffvelcdm 7083	. . . . . . . . . 10 ⊢ ((𝑔:dom (iEdg‘𝐴)⟶dom (iEdg‘𝐵) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐵))
15	14	ex 412	. . . . . . . . 9 ⊢ (𝑔:dom (iEdg‘𝐴)⟶dom (iEdg‘𝐵) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐵)))
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐵)))
17	16	adantr 480	. . . . . . 7 ⊢ ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐵)))
18	17	ad2antlr 724	. . . . . 6 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → (𝑗 ∈ dom (iEdg‘𝐴) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐵)))
19	18	imp 406	. . . . 5 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑔‘𝑗) ∈ dom (iEdg‘𝐵))
20		fveq2 6891	. . . . . . . 8 ⊢ (𝑘 = (𝑔‘𝑗) → ((iEdg‘𝐵)‘𝑘) = ((iEdg‘𝐵)‘(𝑔‘𝑗)))
21	20	imaeq2d 6059	. . . . . . 7 ⊢ (𝑘 = (𝑔‘𝑗) → (𝑣 “ ((iEdg‘𝐵)‘𝑘)) = (𝑣 “ ((iEdg‘𝐵)‘(𝑔‘𝑗))))
22		2fveq3 6896	. . . . . . 7 ⊢ (𝑘 = (𝑔‘𝑗) → ((iEdg‘𝐶)‘(𝑤‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗))))
23	21, 22	eqeq12d 2747	. . . . . 6 ⊢ (𝑘 = (𝑔‘𝑗) → ((𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)) ↔ (𝑣 “ ((iEdg‘𝐵)‘(𝑔‘𝑗))) = ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗)))))
24	23	rspccv 3609	. . . . 5 ⊢ (∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)) → ((𝑔‘𝑗) ∈ dom (iEdg‘𝐵) → (𝑣 “ ((iEdg‘𝐵)‘(𝑔‘𝑗))) = ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗)))))
25	12, 19, 24	sylc 65	. . . 4 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑣 “ ((iEdg‘𝐵)‘(𝑔‘𝑗))) = ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗))))
26	11, 25	eqtrd 2771	. . 3 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑣 “ (𝑓 “ ((iEdg‘𝐴)‘𝑗))) = ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗))))
27		f1ofn 6834	. . . . . . . 8 ⊢ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → 𝑔 Fn dom (iEdg‘𝐴))
28	27	adantr 480	. . . . . . 7 ⊢ ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → 𝑔 Fn dom (iEdg‘𝐴))
29	28	ad2antlr 724	. . . . . 6 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝑔 Fn dom (iEdg‘𝐴))
30		fvco2 6988	. . . . . 6 ⊢ ((𝑔 Fn dom (iEdg‘𝐴) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → ((𝑤 ∘ 𝑔)‘𝑗) = (𝑤‘(𝑔‘𝑗)))
31	29, 30	sylan 579	. . . . 5 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → ((𝑤 ∘ 𝑔)‘𝑗) = (𝑤‘(𝑔‘𝑗)))
32	31	eqcomd 2737	. . . 4 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → (𝑤‘(𝑔‘𝑗)) = ((𝑤 ∘ 𝑔)‘𝑗))
33	32	fveq2d 6895	. . 3 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗))) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))
34	2, 26, 33	3eqtrd 2775	. 2 ⊢ ((((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) ∧ 𝑗 ∈ dom (iEdg‘𝐴)) → ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))
35	34	ralrimiva 3145	1 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  dom cdm 5676   “ cima 5679   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  Vtxcvtx 28690  iEdgciedg 28691  UHGraphcuhgr 28750

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-f1o 6550  df-fv 6551

This theorem is referenced by:  isomgrtr  46968