Theorem funcestrcsetclem9 16788

Description: Lemma 9 for funcestrcsetc 16789. (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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑈 ∈ WUni)
4		eqid 2622	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5	1, 2	estrcbas 16765	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
6		funcestrcsetc.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐸)
7	5, 6	syl6reqr 2675	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = 𝑈)
8	7	eleq2d 2687	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈))
9	8	biimpcd 239	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈))
10	9	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈))
11	10	impcom 446	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝑈)
12	7	eleq2d 2687	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈))
13	12	biimpcd 239	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝜑 → 𝑌 ∈ 𝑈))
14	13	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑌 ∈ 𝑈))
15	14	impcom 446	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝑈)
16		eqid 2622	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
17		eqid 2622	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	1, 3, 4, 11, 15, 16, 17	estrchom 16767	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
19	18	eleq2d 2687	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
20	7	eleq2d 2687	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ 𝑈))
21	20	biimpcd 239	. . . . . . . 8 ⊢ (𝑍 ∈ 𝐵 → (𝜑 → 𝑍 ∈ 𝑈))
22	21	3ad2ant3 1084	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑍 ∈ 𝑈))
23	22	impcom 446	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝑈)
24		eqid 2622	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
25	1, 3, 4, 15, 23, 17, 24	estrchom 16767	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
26	25	eleq2d 2687	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))))
27	19, 26	anbi12d 747	. . 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 7879	. . . . . . . . . 10 ⊢ (𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29		elmapi 7879	. . . . . . . . . 10 ⊢ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30		fco 6058	. . . . . . . . . 10 ⊢ ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
31	28, 29, 30	syl2an 494	. . . . . . . . 9 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32		fvex 6201	. . . . . . . . . 10 ⊢ (Base‘𝑍) ∈ V
33		fvex 6201	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
34	32, 33	elmap 7886	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)) ↔ (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
35	31, 34	sylibr 224	. . . . . . . 8 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
36	35	ancoms 469	. . . . . . 7 ⊢ ((𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
37	36	adantl 482	. . . . . 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 6439	. . . . . 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, 6, 41, 2, 42, 43, 16, 24	funcestrcsetclem5 16784	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
45	44	3adantr2 1221	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
46	45	adantr 481	. . . . . 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 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑈 ∈ WUni)
48		eqid 2622	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
49	11	adantr 481	. . . . . . 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 481	. . . . . . 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 481	. . . . . . 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 764	. . . . . . 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 765	. . . . . . 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 16770	. . . . . 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 6197	. . . . 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 2622	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
57	1, 40, 6, 41, 2, 42	funcestrcsetclem2 16781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
58	57	3ad2antr1 1226	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
59	58	adantr 481	. . . . . . 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, 6, 41, 2, 42	funcestrcsetclem2 16781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
61	60	3ad2antr2 1227	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
62	61	adantr 481	. . . . . . 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, 6, 41, 2, 42	funcestrcsetclem2 16781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ 𝑈)
64	63	3ad2antr3 1228	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) ∈ 𝑈)
65	64	adantr 481	. . . . . . 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, 6, 41, 2, 42	funcestrcsetclem1 16780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
67	66	3ad2antr1 1226	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
68	1, 40, 6, 41, 2, 42	funcestrcsetclem1 16780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
69	68	3ad2antr2 1227	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
70	67, 69	feq23d 6040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
71	70	adantr 481	. . . . . . . . 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 247	. . . . . . . 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 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝜑)
74		3simpa 1058	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
75	74	ad2antlr 763	. . . . . . . . . 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 794	. . . . . . . . . 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, 6, 41, 2, 42, 43, 16, 17	funcestrcsetclem6 16785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
78	73, 75, 76, 77	syl3anc 1326	. . . . . . . . 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 6030	. . . . . . . 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 247	. . . . . . 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, 6, 41, 2, 42	funcestrcsetclem1 16780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) = (Base‘𝑍))
82	81	3ad2antr3 1228	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) = (Base‘𝑍))
83	69, 82	feq23d 6040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
84	83	adantr 481	. . . . . . . . 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 247	. . . . . . . 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 1060	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
87	86	ad2antlr 763	. . . . . . . . . 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 796	. . . . . . . . . 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, 6, 41, 2, 42, 43, 17, 24	funcestrcsetclem6 16785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
90	73, 87, 88, 89	syl3anc 1326	. . . . . . . . 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 6030	. . . . . . . 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 247	. . . . . . 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 16733	. . . . . 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 5286	. . . . . 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 2656	. . . . 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 2666	. . . 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 450	. . 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 230	. 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 1261	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   ↦ cmpt 4729   I cid 5023   ↾ cres 5116   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857  WUnicwun 9522  Basecbs 15857  Hom chom 15952  compcco 15953  SetCatcsetc 16725  ExtStrCatcestrc 16762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wun 9524  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-setc 16726  df-estrc 16763

This theorem is referenced by:  funcestrcsetc  16789