Theorem isomgrtrlem 45178

Description: Lemma for isomgrtr 45179. (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 6144	. . . 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 6756	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((iEdg‘𝐴)‘𝑖) = ((iEdg‘𝐴)‘𝑗))
4	3	imaeq2d 5958	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (𝑓 “ ((iEdg‘𝐴)‘𝑗)))
5		2fveq3 6761	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((iEdg‘𝐵)‘(𝑔‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑗)))
6	4, 5	eqeq12d 2754	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ (𝑓 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐵)‘(𝑔‘𝑗))))
7	6	rspccv 3549	. . . . . . . 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 723	. . . . . 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 5958	. . . 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 774	. . . . 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 6700	. . . . . . . . 9 ⊢ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → 𝑔:dom (iEdg‘𝐴)⟶dom (iEdg‘𝐵))
14		ffvelrn 6941	. . . . . . . . . 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 723	. . . . . 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 6756	. . . . . . . 8 ⊢ (𝑘 = (𝑔‘𝑗) → ((iEdg‘𝐵)‘𝑘) = ((iEdg‘𝐵)‘(𝑔‘𝑗)))
21	20	imaeq2d 5958	. . . . . . 7 ⊢ (𝑘 = (𝑔‘𝑗) → (𝑣 “ ((iEdg‘𝐵)‘𝑘)) = (𝑣 “ ((iEdg‘𝐵)‘(𝑔‘𝑗))))
22		2fveq3 6761	. . . . . . 7 ⊢ (𝑘 = (𝑔‘𝑗) → ((iEdg‘𝐶)‘(𝑤‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘(𝑔‘𝑗))))
23	21, 22	eqeq12d 2754	. . . . . 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 3549	. . . . 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 2778	. . 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 6701	. . . . . . . 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 723	. . . . . 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 6847	. . . . . 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 2744	. . . 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 6760	. . 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 2782	. 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 3107	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 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  dom cdm 5580   “ cima 5583   ∘ ccom 5584   Fn wfn 6413  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  Vtxcvtx 27269  iEdgciedg 27270  UHGraphcuhgr 27329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-f1o 6425  df-fv 6426

This theorem is referenced by:  isomgrtr  45179