Theorem funcestrcsetclem9 17390

Description: Lemma 9 for funcestrcsetc 17391. (Contributed by AV, 23-Mar-2020.)

Hypotheses

Ref	Expression
funcestrcsetc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b	⊢ 𝐵 = (Base‘𝐸)
funcestrcsetc.c	⊢ 𝐶 = (Base‘𝑆)
funcestrcsetc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcestrcsetc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))

Assertion

Ref	Expression
funcestrcsetclem9	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝜑,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦

Allowed substitution hints:   𝐶(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem funcestrcsetclem9

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	. . . . . 6 ⊢ 𝐸 = (ExtStrCat‘𝑈)
2		funcestrcsetc.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ WUni)
3	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑈 ∈ WUni)
4		eqid 2798	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		funcestrcsetc.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐸)
6	1, 2	estrcbas 17367	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
7	5, 6	eqtr4id 2852	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = 𝑈)
8	7	eleq2d 2875	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈))
9	8	biimpcd 252	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈))
10	9	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈))
11	10	impcom 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝑈)
12	7	eleq2d 2875	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈))
13	12	biimpcd 252	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝜑 → 𝑌 ∈ 𝑈))
14	13	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑌 ∈ 𝑈))
15	14	impcom 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝑈)
16		eqid 2798	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
17		eqid 2798	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	1, 3, 4, 11, 15, 16, 17	estrchom 17369	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
19	18	eleq2d 2875	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
20	7	eleq2d 2875	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ 𝑈))
21	20	biimpcd 252	. . . . . . . 8 ⊢ (𝑍 ∈ 𝐵 → (𝜑 → 𝑍 ∈ 𝑈))
22	21	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑍 ∈ 𝑈))
23	22	impcom 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝑈)
24		eqid 2798	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
25	1, 3, 4, 15, 23, 17, 24	estrchom 17369	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑m (Base‘𝑌)))
26	25	eleq2d 2875	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))))
27	19, 26	anbi12d 633	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍)) ↔ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))))
28		elmapi 8411	. . . . . . . . . 10 ⊢ (𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29		elmapi 8411	. . . . . . . . . 10 ⊢ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30		fco 6505	. . . . . . . . . 10 ⊢ ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
31	28, 29, 30	syl2an 598	. . . . . . . . 9 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32		fvex 6658	. . . . . . . . . 10 ⊢ (Base‘𝑍) ∈ V
33		fvex 6658	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
34	32, 33	elmap 8418	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) ↔ (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
35	31, 34	sylibr 237	. . . . . . . 8 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
36	35	ancoms 462	. . . . . . 7 ⊢ ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
37	36	adantl 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
38		fvresi 6912	. . . . . 6 ⊢ ((𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) → (( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋)))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
39	37, 38	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋)))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
40		funcestrcsetc.s	. . . . . . . . 9 ⊢ 𝑆 = (SetCat‘𝑈)
41		funcestrcsetc.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝑆)
42		funcestrcsetc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
43		funcestrcsetc.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
44	1, 40, 5, 41, 2, 42, 43, 16, 24	funcestrcsetclem5 17386	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
45	44	3adantr2 1167	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
46	45	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
47	3	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑈 ∈ WUni)
48		eqid 2798	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
49	11	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑋 ∈ 𝑈)
50	15	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑌 ∈ 𝑈)
51	23	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑍 ∈ 𝑈)
52	29	ad2antrl 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
53	28	ad2antll 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
54	1, 47, 48, 49, 50, 51, 16, 17, 24, 52, 53	estrcco 17372	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻) = (𝐾 ∘ 𝐻))
55	46, 54	fveq12d 6652	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋)))‘(𝐾 ∘ 𝐻)))
56		eqid 2798	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
57	1, 40, 5, 41, 2, 42	funcestrcsetclem2 17383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
58	57	3ad2antr1 1185	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
59	58	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹‘𝑋) ∈ 𝑈)
60	1, 40, 5, 41, 2, 42	funcestrcsetclem2 17383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
61	60	3ad2antr2 1186	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
62	61	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹‘𝑌) ∈ 𝑈)
63	1, 40, 5, 41, 2, 42	funcestrcsetclem2 17383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ 𝑈)
64	63	3ad2antr3 1187	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) ∈ 𝑈)
65	64	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹‘𝑍) ∈ 𝑈)
66	1, 40, 5, 41, 2, 42	funcestrcsetclem1 17382	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
67	66	3ad2antr1 1185	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
68	1, 40, 5, 41, 2, 42	funcestrcsetclem1 17382	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
69	68	3ad2antr2 1186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
70	67, 69	feq23d 6482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
71	70	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
72	52, 71	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌))
73		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝜑)
74		3simpa 1145	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
75	74	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
76		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
77	1, 40, 5, 41, 2, 42, 43, 16, 17	funcestrcsetclem6 17387	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
78	73, 75, 76, 77	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
79	78	feq1d 6472	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌)))
80	72, 79	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻):(𝐹‘𝑋)⟶(𝐹‘𝑌))
81	1, 40, 5, 41, 2, 42	funcestrcsetclem1 17382	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) = (Base‘𝑍))
82	81	3ad2antr3 1187	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) = (Base‘𝑍))
83	69, 82	feq23d 6482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
84	83	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
85	53, 84	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍))
86		3simpc 1147	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
87	86	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
88		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))
89	1, 40, 5, 41, 2, 42, 43, 17, 24	funcestrcsetclem6 17387	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
90	73, 87, 88, 89	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
91	90	feq1d 6472	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍)))
92	85, 91	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾):(𝐹‘𝑌)⟶(𝐹‘𝑍))
93	40, 47, 56, 59, 62, 65, 80, 92	setcco 17335	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
94	90, 78	coeq12d 5699	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾 ∘ 𝐻))
95	93, 94	eqtrd 2833	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾 ∘ 𝐻))
96	39, 55, 95	3eqtr4d 2843	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
97	96	ex 416	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))))
98	27, 97	sylbid 243	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))))
99	98	3impia 1114	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   ↦ cmpt 5110   I cid 5424   ↾ cres 5521   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389  WUnicwun 10111  Basecbs 16475  Hom chom 16568  compcco 16569  SetCatcsetc 17327  ExtStrCatcestrc 17364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-wun 10113  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-hom 16581  df-cco 16582  df-setc 17328  df-estrc 17365

This theorem is referenced by:  funcestrcsetc  17391