Theorem funcestrcsetclem9 18322

Description: Lemma 9 for funcestrcsetc 18323. (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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑈 ∈ WUni)
4		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5		funcestrcsetc.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐸)
6	1, 2	estrcbas 18299	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
7	5, 6	eqtr4id 2815	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = 𝑈)
8	7	eleq2d 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈))
9	8	biimpcd 252	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈))
10	9	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈))
11	10	impcom 413	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝑈)
12	7	eleq2d 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈))
13	12	biimpcd 252	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝜑 → 𝑌 ∈ 𝑈))
14	13	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑌 ∈ 𝑈))
15	14	impcom 413	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝑈)
16		eqid 2761	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
17		eqid 2761	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	1, 3, 4, 11, 15, 16, 17	estrchom 18301	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
19	18	eleq2d 2847	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
20	7	eleq2d 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ 𝑈))
21	20	biimpcd 252	. . . . . . . 8 ⊢ (𝑍 ∈ 𝐵 → (𝜑 → 𝑍 ∈ 𝑈))
22	21	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝜑 → 𝑍 ∈ 𝑈))
23	22	impcom 413	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝑈)
24		eqid 2761	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
25	1, 3, 4, 15, 23, 17, 24	estrchom 18301	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑m (Base‘𝑌)))
26	25	eleq2d 2847	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))))
27	19, 26	anbi12d 644	. . 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 8869	. . . . . . . . . 10 ⊢ (𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29		elmapi 8869	. . . . . . . . . 10 ⊢ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30		fco 6734	. . . . . . . . . 10 ⊢ ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
31	28, 29, 30	syl2an 608	. . . . . . . . 9 ⊢ ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾 ∘ 𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32		fvex 6898	. . . . . . . . . 10 ⊢ (Base‘𝑍) ∈ V
33		fvex 6898	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
34	32, 33	elmap 8899	. . . . . . . . 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 464	. . . . . . 7 ⊢ ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → (𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
37	36	adantl 487	. . . . . 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 7178	. . . . . 6 ⊢ ((𝐾 ∘ 𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) → (( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋)))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
39	37, 38	syl 18	. . . . 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 18318	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
45	44	3adantr2 1189	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
46	45	adantr 486	. . . . . 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 486	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑈 ∈ WUni)
48		eqid 2761	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
49	11	adantr 486	. . . . . . 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 486	. . . . . . 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 486	. . . . . . 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 741	. . . . . . 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 742	. . . . . . 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 18304	. . . . . 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 6892	. . . . 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 2761	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
57	1, 40, 5, 41, 2, 42	funcestrcsetclem2 18315	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
58	57	3ad2antr1 1207	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
59	58	adantr 486	. . . . . . 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 18315	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
61	60	3ad2antr2 1208	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
62	61	adantr 486	. . . . . . 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 18315	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ 𝑈)
64	63	3ad2antr3 1209	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) ∈ 𝑈)
65	64	adantr 486	. . . . . . 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 18314	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
67	66	3ad2antr1 1207	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
68	1, 40, 5, 41, 2, 42	funcestrcsetclem1 18314	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
69	68	3ad2antr2 1208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
70	67, 69	feq23d 6704	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
71	70	adantr 486	. . . . . . . . 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 779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝜑)
74		3simpa 1166	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
75	74	ad2antlr 740	. . . . . . . . . 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 783	. . . . . . . . . 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 18319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
78	73, 75, 76, 77	syl3anc 1398	. . . . . . . . 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 6691	. . . . . . . 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 18314	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) = (Base‘𝑍))
82	81	3ad2antr3 1209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) = (Base‘𝑍))
83	69, 82	feq23d 6704	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
84	83	adantr 486	. . . . . . . . 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 1168	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
87	86	ad2antlr 740	. . . . . . . . . 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 785	. . . . . . . . . 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 18319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
90	73, 87, 88, 89	syl3anc 1398	. . . . . . . . 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 6691	. . . . . . . 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 18258	. . . . . 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 5842	. . . . . 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 2796	. . . . 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 2806	. . . 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 418	. . 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 1135	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   ↦ cmpt 5186   I cid 5545   ↾ cres 5653   ∘ ccom 5655  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ↑_m cmap 8847  WUnicwun 10785  Basecbs 17387  Hom chom 17439  compcco 17440  SetCatcsetc 18250  ExtStrCatcestrc 18296

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-wun 10787  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-setc 18251  df-estrc 18297

This theorem is used by:  funcestrcsetc  18323