Theorem funcestrcsetclem9 17893

Description: Lemma 9 for funcestrcsetc 17894. (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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑈 ∈ WUni)
4		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		funcestrcsetc.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐸)
6	1, 2	estrcbas 17869	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
7	5, 6	eqtr4id 2792	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = 𝑈)
8	7	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈))
9	8	biimpcd 248	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈))
10	9	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈))
11	10	impcom 407	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝑈)
12	7	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈))
13	12	biimpcd 248	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝜑 → 𝑌 ∈ 𝑈))
14	13	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑌 ∈ 𝑈))
15	14	impcom 407	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝑈)
16		eqid 2733	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
17		eqid 2733	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	1, 3, 4, 11, 15, 16, 17	estrchom 17871	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
19	18	eleq2d 2819	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
20	7	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ 𝑈))
21	20	biimpcd 248	. . . . . . . 8 ⊢ (𝑍 ∈ 𝐵 → (𝜑 → 𝑍 ∈ 𝑈))
22	21	3ad2ant3 1133	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑍 ∈ 𝑈))
23	22	impcom 407	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝑈)
24		eqid 2733	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
25	1, 3, 4, 15, 23, 17, 24	estrchom 17871	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑m (Base‘𝑌)))
26	25	eleq2d 2819	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))))
27	19, 26	anbi12d 630	. . 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 8657	. . . . . . . . . 10 ⊢ (𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29		elmapi 8657	. . . . . . . . . 10 ⊢ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30		fco 6642	. . . . . . . . . 10 ⊢ ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
31	28, 29, 30	syl2an 595	. . . . . . . . 9 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32		fvex 6805	. . . . . . . . . 10 ⊢ (Base‘𝑍) ∈ V
33		fvex 6805	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
34	32, 33	elmap 8679	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) ↔ (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
35	31, 34	sylibr 233	. . . . . . . 8 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
36	35	ancoms 458	. . . . . . 7 ⊢ ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
37	36	adantl 481	. . . . . 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 7065	. . . . . 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 17889	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
45	44	3adantr2 1168	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
46	45	adantr 480	. . . . . 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 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑈 ∈ WUni)
48		eqid 2733	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
49	11	adantr 480	. . . . . . 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 480	. . . . . . 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 480	. . . . . . 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 724	. . . . . . 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 725	. . . . . . 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 17874	. . . . . 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 6799	. . . . 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 2733	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
57	1, 40, 5, 41, 2, 42	funcestrcsetclem2 17886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
58	57	3ad2antr1 1186	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
59	58	adantr 480	. . . . . . 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 17886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
61	60	3ad2antr2 1187	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
62	61	adantr 480	. . . . . . 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 17886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ 𝑈)
64	63	3ad2antr3 1188	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) ∈ 𝑈)
65	64	adantr 480	. . . . . . 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 17885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
67	66	3ad2antr1 1186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
68	1, 40, 5, 41, 2, 42	funcestrcsetclem1 17885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
69	68	3ad2antr2 1187	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
70	67, 69	feq23d 6613	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
71	70	adantr 480	. . . . . . . . 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 256	. . . . . . . 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 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝜑)
74		3simpa 1146	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
75	74	ad2antlr 723	. . . . . . . . . 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 767	. . . . . . . . . 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 17890	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
78	73, 75, 76, 77	syl3anc 1369	. . . . . . . . 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 6603	. . . . . . . 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 256	. . . . . . 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 17885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) = (Base‘𝑍))
82	81	3ad2antr3 1188	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) = (Base‘𝑍))
83	69, 82	feq23d 6613	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
84	83	adantr 480	. . . . . . . . 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 256	. . . . . . . 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 1148	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
87	86	ad2antlr 723	. . . . . . . . . 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 769	. . . . . . . . . 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 17890	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
90	73, 87, 88, 89	syl3anc 1369	. . . . . . . . 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 6603	. . . . . . . 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 256	. . . . . . 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 17826	. . . . . 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 5777	. . . . . 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 2773	. . . . 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 2783	. . . 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 412	. . 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 239	. 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 1115	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1537   ∈ wcel 2101  ⟨cop 4570   ↦ cmpt 5160   I cid 5490   ↾ cres 5593   ∘ ccom 5595  ⟶wf 6443  ‘cfv 6447  (class class class)co 7295   ∈ cmpo 7297   ↑_m cmap 8635  WUnicwun 10484  Basecbs 16940  Hom chom 17001  compcco 17002  SetCatcsetc 17818  ExtStrCatcestrc 17866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-cnex 10955  ax-resscn 10956  ax-1cn 10957  ax-icn 10958  ax-addcl 10959  ax-addrcl 10960  ax-mulcl 10961  ax-mulrcl 10962  ax-mulcom 10963  ax-addass 10964  ax-mulass 10965  ax-distr 10966  ax-i2m1 10967  ax-1ne0 10968  ax-1rid 10969  ax-rnegex 10970  ax-rrecex 10971  ax-cnre 10972  ax-pre-lttri 10973  ax-pre-lttrn 10974  ax-pre-ltadd 10975  ax-pre-mulgt0 10976

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-om 7733  df-1st 7851  df-2nd 7852  df-frecs 8117  df-wrecs 8148  df-recs 8222  df-rdg 8261  df-1o 8317  df-er 8518  df-map 8637  df-en 8754  df-dom 8755  df-sdom 8756  df-fin 8757  df-wun 10486  df-pnf 11039  df-mnf 11040  df-xr 11041  df-ltxr 11042  df-le 11043  df-sub 11235  df-neg 11236  df-nn 12002  df-2 12064  df-3 12065  df-4 12066  df-5 12067  df-6 12068  df-7 12069  df-8 12070  df-9 12071  df-n0 12262  df-z 12348  df-dec 12466  df-uz 12611  df-fz 13268  df-struct 16876  df-slot 16911  df-ndx 16923  df-base 16941  df-hom 17014  df-cco 17015  df-setc 17819  df-estrc 17867

This theorem is referenced by:  funcestrcsetc  17894