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Theorem funcsetcestrclem9 18111

Description: Lemma 9 for funcsetcestrc 18112. (Contributed by AV, 28-Mar-2020.)

**Hypotheses**
Ref	Expression
funcsetcestrc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c	⊢ 𝐶 = (Base‘𝑆)
funcsetcestrc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcsetcestrc.o	⊢ (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))
funcsetcestrc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)

**Assertion**
Ref	Expression
funcsetcestrclem9	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))

Distinct variable groups: 𝑥,𝐶 𝑥,𝑋 𝜑,𝑥 𝑦,𝐶,𝑥 𝑦,𝑋 𝑥,𝑌,𝑦 𝜑,𝑦 𝑥,𝑍,𝑦 Allowed substitution hints: 𝑆(𝑥,𝑦) 𝑈(𝑥,𝑦) 𝐸(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝐻(𝑥,𝑦) 𝐾(𝑥,𝑦)

**Proof of Theorem funcsetcestrclem9**
Step	Hyp	Ref	Expression
1		funcsetcestrc.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
2		funcsetcestrc.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ WUni)
3	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → 𝑈 ∈ WUni)
4		eqid 2732	. . . . . 6 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
5		funcsetcestrc.c	. . . . . . . . . . 11 ⊢ 𝐶 = (Base‘𝑆)
6	1, 2	setcbas 18024	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
7	5, 6	eqtr4id 2791	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = 𝑈)
8	7	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈))
9	8	biimpcd 248	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → (𝜑 → 𝑋 ∈ 𝑈))
10	9	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (𝜑 → 𝑋 ∈ 𝑈))
11	10	impcom 408	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → 𝑋 ∈ 𝑈)
12	7	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐶 ↔ 𝑌 ∈ 𝑈))
13	12	biimpcd 248	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐶 → (𝜑 → 𝑌 ∈ 𝑈))
14	13	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (𝜑 → 𝑌 ∈ 𝑈))
15	14	impcom 408	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → 𝑌 ∈ 𝑈)
16	1, 3, 4, 11, 15	setchom 18026	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝑋(Hom ‘𝑆)𝑌) = (𝑌 ↑_m 𝑋))
17	16	eleq2d 2819	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ↔ 𝐻 ∈ (𝑌 ↑_m 𝑋)))
18	7	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ 𝐶 ↔ 𝑍 ∈ 𝑈))
19	18	biimpcd 248	. . . . . . . 8 ⊢ (𝑍 ∈ 𝐶 → (𝜑 → 𝑍 ∈ 𝑈))
20	19	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (𝜑 → 𝑍 ∈ 𝑈))
21	20	impcom 408	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → 𝑍 ∈ 𝑈)
22	1, 3, 4, 15, 21	setchom 18026	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝑌(Hom ‘𝑆)𝑍) = (𝑍 ↑_m 𝑌))
23	22	eleq2d 2819	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍) ↔ 𝐾 ∈ (𝑍 ↑_m 𝑌)))
24	17, 23	anbi12d 631	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) ↔ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))))
25		elmapi 8839	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑍 ↑_m 𝑌) → 𝐾:𝑌⟶𝑍)
26		elmapi 8839	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑌 ↑_m 𝑋) → 𝐻:𝑋⟶𝑌)
27		fco 6738	. . . . . . . . 9 ⊢ ((𝐾:𝑌⟶𝑍 ∧ 𝐻:𝑋⟶𝑌) → (𝐾 ∘ 𝐻):𝑋⟶𝑍)
28	25, 26, 27	syl2anr 597	. . . . . . . 8 ⊢ ((𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌)) → (𝐾 ∘ 𝐻):𝑋⟶𝑍)
29	28	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝐾 ∘ 𝐻):𝑋⟶𝑍)
30		elmapg 8829	. . . . . . . . . 10 ⊢ ((𝑍 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶) → ((𝐾 ∘ 𝐻) ∈ (𝑍 ↑_m 𝑋) ↔ (𝐾 ∘ 𝐻):𝑋⟶𝑍))
31	30	ancoms 459	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → ((𝐾 ∘ 𝐻) ∈ (𝑍 ↑_m 𝑋) ↔ (𝐾 ∘ 𝐻):𝑋⟶𝑍))
32	31	3adant2 1131	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → ((𝐾 ∘ 𝐻) ∈ (𝑍 ↑_m 𝑋) ↔ (𝐾 ∘ 𝐻):𝑋⟶𝑍))
33	32	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝐾 ∘ 𝐻) ∈ (𝑍 ↑_m 𝑋) ↔ (𝐾 ∘ 𝐻):𝑋⟶𝑍))
34	29, 33	mpbird 256	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝐾 ∘ 𝐻) ∈ (𝑍 ↑_m 𝑋))
35		fvresi 7167	. . . . . 6 ⊢ ((𝐾 ∘ 𝐻) ∈ (𝑍 ↑_m 𝑋) → (( I ↾ (𝑍 ↑_m 𝑋))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
36	34, 35	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (( I ↾ (𝑍 ↑_m 𝑋))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
37		funcsetcestrc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
38		funcsetcestrc.o	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ 𝑈)
39		funcsetcestrc.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))
40	1, 5, 37, 2, 38, 39	funcsetcestrclem5 18107	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍 ↑_m 𝑋)))
41	40	3adantr2 1170	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍 ↑_m 𝑋)))
42	41	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝑋𝐺𝑍) = ( I ↾ (𝑍 ↑_m 𝑋)))
43	3	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝑈 ∈ WUni)
44		eqid 2732	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
45	11	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝑋 ∈ 𝑈)
46	15	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝑌 ∈ 𝑈)
47	21	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝑍 ∈ 𝑈)
48	26	ad2antrl 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝐻:𝑋⟶𝑌)
49	25	ad2antll 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝐾:𝑌⟶𝑍)
50	1, 43, 44, 45, 46, 47, 48, 49	setcco 18029	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻) = (𝐾 ∘ 𝐻))
51	42, 50	fveq12d 6895	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (( I ↾ (𝑍 ↑_m 𝑋))‘(𝐾 ∘ 𝐻)))
52		funcsetcestrc.e	. . . . . . 7 ⊢ 𝐸 = (ExtStrCat‘𝑈)
53		eqid 2732	. . . . . . 7 ⊢ (comp‘𝐸) = (comp‘𝐸)
54	1, 5, 37, 2, 38	funcsetcestrclem2 18103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ 𝑈)
55	54	3ad2antr1 1188	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐹‘𝑋) ∈ 𝑈)
56	55	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝐹‘𝑋) ∈ 𝑈)
57	1, 5, 37, 2, 38	funcsetcestrclem2 18103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐶) → (𝐹‘𝑌) ∈ 𝑈)
58	57	3ad2antr2 1189	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐹‘𝑌) ∈ 𝑈)
59	58	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝐹‘𝑌) ∈ 𝑈)
60	1, 5, 37, 2, 38	funcsetcestrclem2 18103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐶) → (𝐹‘𝑍) ∈ 𝑈)
61	60	3ad2antr3 1190	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐹‘𝑍) ∈ 𝑈)
62	61	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝐹‘𝑍) ∈ 𝑈)
63		eqid 2732	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑋)) = (Base‘(𝐹‘𝑋))
64		eqid 2732	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑌)) = (Base‘(𝐹‘𝑌))
65		eqid 2732	. . . . . . 7 ⊢ (Base‘(𝐹‘𝑍)) = (Base‘(𝐹‘𝑍))
66		simpll 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝜑)
67		3simpa 1148	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶))
68	67	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶))
69		simprl 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝐻 ∈ (𝑌 ↑_m 𝑋))
70	1, 5, 37, 2, 38, 39	funcsetcestrclem6 18108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑_m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
71	66, 68, 69, 70	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
72	1, 5, 37	funcsetcestrclem1 18102	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})
73	72	3ad2antr1 1188	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})
74	73	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘(𝐹‘𝑋)) = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
75		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ {⟨(Base‘ndx), 𝑋⟩} = {⟨(Base‘ndx), 𝑋⟩}
76	75	1strbas 17157	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐶 → 𝑋 = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
77	76	eqcomd 2738	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐶 → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
78	77	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
79	78	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
80	74, 79	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘(𝐹‘𝑋)) = 𝑋)
81	80	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (Base‘(𝐹‘𝑋)) = 𝑋)
82	1, 5, 37	funcsetcestrclem1 18102	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐶) → (𝐹‘𝑌) = {⟨(Base‘ndx), 𝑌⟩})
83	82	3ad2antr2 1189	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐹‘𝑌) = {⟨(Base‘ndx), 𝑌⟩})
84	83	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘(𝐹‘𝑌)) = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
85		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ {⟨(Base‘ndx), 𝑌⟩} = {⟨(Base‘ndx), 𝑌⟩}
86	85	1strbas 17157	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ 𝐶 → 𝑌 = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
87	86	eqcomd 2738	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ 𝐶 → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
88	87	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
89	88	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
90	84, 89	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘(𝐹‘𝑌)) = 𝑌)
91	90	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (Base‘(𝐹‘𝑌)) = 𝑌)
92	71, 81, 91	feq123d 6703	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹‘𝑋))⟶(Base‘(𝐹‘𝑌)) ↔ 𝐻:𝑋⟶𝑌))
93	48, 92	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹‘𝑋))⟶(Base‘(𝐹‘𝑌)))
94		3simpc 1150	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶))
95	94	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶))
96		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → 𝐾 ∈ (𝑍 ↑_m 𝑌))
97	1, 5, 37, 2, 38, 39	funcsetcestrclem6 18108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
98	66, 95, 96, 97	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
99	1, 5, 37	funcsetcestrclem1 18102	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐶) → (𝐹‘𝑍) = {⟨(Base‘ndx), 𝑍⟩})
100	99	3ad2antr3 1190	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (𝐹‘𝑍) = {⟨(Base‘ndx), 𝑍⟩})
101	100	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘(𝐹‘𝑍)) = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
102		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ {⟨(Base‘ndx), 𝑍⟩} = {⟨(Base‘ndx), 𝑍⟩}
103	102	1strbas 17157	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝐶 → 𝑍 = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
104	103	eqcomd 2738	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ 𝐶 → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
105	104	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
106	105	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
107	101, 106	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → (Base‘(𝐹‘𝑍)) = 𝑍)
108	107	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (Base‘(𝐹‘𝑍)) = 𝑍)
109	98, 91, 108	feq123d 6703	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹‘𝑌))⟶(Base‘(𝐹‘𝑍)) ↔ 𝐾:𝑌⟶𝑍))
110	49, 109	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹‘𝑌))⟶(Base‘(𝐹‘𝑍)))
111	52, 43, 53, 56, 59, 62, 63, 64, 65, 93, 110	estrcco 18077	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
112	98, 71	coeq12d 5862	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾 ∘ 𝐻))
113	111, 112	eqtrd 2772	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾 ∘ 𝐻))
114	36, 51, 113	3eqtr4d 2782	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) ∧ (𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
115	114	ex 413	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → ((𝐻 ∈ (𝑌 ↑_m 𝑋) ∧ 𝐾 ∈ (𝑍 ↑_m 𝑌)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))))
116	24, 115	sylbid 239	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))))
117	116	3impia 1117	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {csn 4627 ⟨cop 4633 ↦ cmpt 5230 I cid 5572 ↾ cres 5677 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 ωcom 7851 ↑_m cmap 8816 WUnicwun 10691 ndxcnx 17122 Basecbs 17140 Hom chom 17204 compcco 17205 SetCatcsetc 18021 ExtStrCatcestrc 18069

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-wun 10693 df-ni 10863 df-pli 10864 df-mi 10865 df-lti 10866 df-plpq 10899 df-mpq 10900 df-ltpq 10901 df-enq 10902 df-nq 10903 df-erq 10904 df-plq 10905 df-mq 10906 df-1nq 10907 df-rq 10908 df-ltnq 10909 df-np 10972 df-plp 10974 df-ltp 10976 df-enr 11046 df-nr 11047 df-c 11112 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-hom 17217 df-cco 17218 df-setc 18022 df-estrc 18070

This theorem is referenced by: funcsetcestrc 18112

W3C validator