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Theorem funcsetcestrclem9 17412
 Description: Lemma 9 for funcsetcestrc 17413. (Contributed by AV, 28-Mar-2020.)
Hypotheses
Ref Expression
funcsetcestrc.s 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c 𝐶 = (Base‘𝑆)
funcsetcestrc.f (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u (𝜑𝑈 ∈ WUni)
funcsetcestrc.o (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
funcsetcestrc.e 𝐸 = (ExtStrCat‘𝑈)
Assertion
Ref Expression
funcsetcestrclem9 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((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
StepHypRef Expression
1 funcsetcestrc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
2 funcsetcestrc.u . . . . . . 7 (𝜑𝑈 ∈ WUni)
32adantr 484 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑈 ∈ WUni)
4 eqid 2798 . . . . . 6 (Hom ‘𝑆) = (Hom ‘𝑆)
5 funcsetcestrc.c . . . . . . . . . . 11 𝐶 = (Base‘𝑆)
61, 2setcbas 17337 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝑆))
75, 6eqtr4id 2852 . . . . . . . . . 10 (𝜑𝐶 = 𝑈)
87eleq2d 2875 . . . . . . . . 9 (𝜑 → (𝑋𝐶𝑋𝑈))
98biimpcd 252 . . . . . . . 8 (𝑋𝐶 → (𝜑𝑋𝑈))
1093ad2ant1 1130 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑋𝑈))
1110impcom 411 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑋𝑈)
127eleq2d 2875 . . . . . . . . 9 (𝜑 → (𝑌𝐶𝑌𝑈))
1312biimpcd 252 . . . . . . . 8 (𝑌𝐶 → (𝜑𝑌𝑈))
14133ad2ant2 1131 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑌𝑈))
1514impcom 411 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑌𝑈)
161, 3, 4, 11, 15setchom 17339 . . . . 5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑋(Hom ‘𝑆)𝑌) = (𝑌m 𝑋))
1716eleq2d 2875 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ↔ 𝐻 ∈ (𝑌m 𝑋)))
187eleq2d 2875 . . . . . . . . 9 (𝜑 → (𝑍𝐶𝑍𝑈))
1918biimpcd 252 . . . . . . . 8 (𝑍𝐶 → (𝜑𝑍𝑈))
20193ad2ant3 1132 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑍𝑈))
2120impcom 411 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑍𝑈)
221, 3, 4, 15, 21setchom 17339 . . . . 5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑌(Hom ‘𝑆)𝑍) = (𝑍m 𝑌))
2322eleq2d 2875 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍) ↔ 𝐾 ∈ (𝑍m 𝑌)))
2417, 23anbi12d 633 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) ↔ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))))
25 elmapi 8418 . . . . . . . . 9 (𝐾 ∈ (𝑍m 𝑌) → 𝐾:𝑌𝑍)
26 elmapi 8418 . . . . . . . . 9 (𝐻 ∈ (𝑌m 𝑋) → 𝐻:𝑋𝑌)
27 fco 6508 . . . . . . . . 9 ((𝐾:𝑌𝑍𝐻:𝑋𝑌) → (𝐾𝐻):𝑋𝑍)
2825, 26, 27syl2anr 599 . . . . . . . 8 ((𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌)) → (𝐾𝐻):𝑋𝑍)
2928adantl 485 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐾𝐻):𝑋𝑍)
30 elmapg 8409 . . . . . . . . . 10 ((𝑍𝐶𝑋𝐶) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3130ancoms 462 . . . . . . . . 9 ((𝑋𝐶𝑍𝐶) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
32313adant2 1128 . . . . . . . 8 ((𝑋𝐶𝑌𝐶𝑍𝐶) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3332ad2antlr 726 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3429, 33mpbird 260 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐾𝐻) ∈ (𝑍m 𝑋))
35 fvresi 6917 . . . . . 6 ((𝐾𝐻) ∈ (𝑍m 𝑋) → (( I ↾ (𝑍m 𝑋))‘(𝐾𝐻)) = (𝐾𝐻))
3634, 35syl 17 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (( I ↾ (𝑍m 𝑋))‘(𝐾𝐻)) = (𝐾𝐻))
37 funcsetcestrc.f . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
38 funcsetcestrc.o . . . . . . . . 9 (𝜑 → ω ∈ 𝑈)
39 funcsetcestrc.g . . . . . . . . 9 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
401, 5, 37, 2, 38, 39funcsetcestrclem5 17408 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑍𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍m 𝑋)))
41403adantr2 1167 . . . . . . 7 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍m 𝑋)))
4241adantr 484 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝑋𝐺𝑍) = ( I ↾ (𝑍m 𝑋)))
433adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑈 ∈ WUni)
44 eqid 2798 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
4511adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑋𝑈)
4615adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑌𝑈)
4721adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑍𝑈)
4826ad2antrl 727 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐻:𝑋𝑌)
4925ad2antll 728 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐾:𝑌𝑍)
501, 43, 44, 45, 46, 47, 48, 49setcco 17342 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻) = (𝐾𝐻))
5142, 50fveq12d 6657 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (( I ↾ (𝑍m 𝑋))‘(𝐾𝐻)))
52 funcsetcestrc.e . . . . . . 7 𝐸 = (ExtStrCat‘𝑈)
53 eqid 2798 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
541, 5, 37, 2, 38funcsetcestrclem2 17404 . . . . . . . . 9 ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ 𝑈)
55543ad2antr1 1185 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑋) ∈ 𝑈)
5655adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐹𝑋) ∈ 𝑈)
571, 5, 37, 2, 38funcsetcestrclem2 17404 . . . . . . . . 9 ((𝜑𝑌𝐶) → (𝐹𝑌) ∈ 𝑈)
58573ad2antr2 1186 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑌) ∈ 𝑈)
5958adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐹𝑌) ∈ 𝑈)
601, 5, 37, 2, 38funcsetcestrclem2 17404 . . . . . . . . 9 ((𝜑𝑍𝐶) → (𝐹𝑍) ∈ 𝑈)
61603ad2antr3 1187 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑍) ∈ 𝑈)
6261adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐹𝑍) ∈ 𝑈)
63 eqid 2798 . . . . . . 7 (Base‘(𝐹𝑋)) = (Base‘(𝐹𝑋))
64 eqid 2798 . . . . . . 7 (Base‘(𝐹𝑌)) = (Base‘(𝐹𝑌))
65 eqid 2798 . . . . . . 7 (Base‘(𝐹𝑍)) = (Base‘(𝐹𝑍))
66 simpll 766 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝜑)
67 3simpa 1145 . . . . . . . . . . 11 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝑋𝐶𝑌𝐶))
6867ad2antlr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝑋𝐶𝑌𝐶))
69 simprl 770 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐻 ∈ (𝑌m 𝑋))
701, 5, 37, 2, 38, 39funcsetcestrclem6 17409 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7166, 68, 69, 70syl3anc 1368 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
721, 5, 37funcsetcestrclem1 17403 . . . . . . . . . . . . 13 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
73723ad2antr1 1185 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
7473fveq2d 6654 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑋)) = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
75 eqid 2798 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑋⟩} = {⟨(Base‘ndx), 𝑋⟩}
76751strbas 16598 . . . . . . . . . . . . . 14 (𝑋𝐶𝑋 = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
7776eqcomd 2804 . . . . . . . . . . . . 13 (𝑋𝐶 → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
78773ad2ant1 1130 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
7978adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
8074, 79eqtrd 2833 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑋)) = 𝑋)
8180adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (Base‘(𝐹𝑋)) = 𝑋)
821, 5, 37funcsetcestrclem1 17403 . . . . . . . . . . . . 13 ((𝜑𝑌𝐶) → (𝐹𝑌) = {⟨(Base‘ndx), 𝑌⟩})
83823ad2antr2 1186 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑌) = {⟨(Base‘ndx), 𝑌⟩})
8483fveq2d 6654 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑌)) = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
85 eqid 2798 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑌⟩} = {⟨(Base‘ndx), 𝑌⟩}
86851strbas 16598 . . . . . . . . . . . . . 14 (𝑌𝐶𝑌 = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
8786eqcomd 2804 . . . . . . . . . . . . 13 (𝑌𝐶 → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
88873ad2ant2 1131 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
8988adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
9084, 89eqtrd 2833 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑌)) = 𝑌)
9190adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (Base‘(𝐹𝑌)) = 𝑌)
9271, 81, 91feq123d 6479 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹𝑋))⟶(Base‘(𝐹𝑌)) ↔ 𝐻:𝑋𝑌))
9348, 92mpbird 260 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹𝑋))⟶(Base‘(𝐹𝑌)))
94 3simpc 1147 . . . . . . . . . . 11 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝑌𝐶𝑍𝐶))
9594ad2antlr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝑌𝐶𝑍𝐶))
96 simprr 772 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐾 ∈ (𝑍m 𝑌))
971, 5, 37, 2, 38, 39funcsetcestrclem6 17409 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐶𝑍𝐶) ∧ 𝐾 ∈ (𝑍m 𝑌)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9866, 95, 96, 97syl3anc 1368 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
991, 5, 37funcsetcestrclem1 17403 . . . . . . . . . . . . 13 ((𝜑𝑍𝐶) → (𝐹𝑍) = {⟨(Base‘ndx), 𝑍⟩})
100993ad2antr3 1187 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑍) = {⟨(Base‘ndx), 𝑍⟩})
101100fveq2d 6654 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑍)) = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
102 eqid 2798 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑍⟩} = {⟨(Base‘ndx), 𝑍⟩}
1031021strbas 16598 . . . . . . . . . . . . . 14 (𝑍𝐶𝑍 = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
104103eqcomd 2804 . . . . . . . . . . . . 13 (𝑍𝐶 → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
1051043ad2ant3 1132 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
106105adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
107101, 106eqtrd 2833 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑍)) = 𝑍)
108107adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (Base‘(𝐹𝑍)) = 𝑍)
10998, 91, 108feq123d 6479 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹𝑌))⟶(Base‘(𝐹𝑍)) ↔ 𝐾:𝑌𝑍))
11049, 109mpbird 260 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹𝑌))⟶(Base‘(𝐹𝑍)))
11152, 43, 53, 56, 59, 62, 63, 64, 65, 93, 110estrcco 17379 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
11298, 71coeq12d 5700 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
113111, 112eqtrd 2833 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
11436, 51, 1133eqtr4d 2843 . . . 4 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
115114ex 416 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
11624, 115sylbid 243 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
1171163impia 1114 1 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {csn 4525  ⟨cop 4531   ↦ cmpt 5111   I cid 5425   ↾ cres 5522   ∘ ccom 5524  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140   ∈ cmpo 7142  ωcom 7567   ↑m cmap 8396  WUnicwun 10118  ndxcnx 16479  Basecbs 16482  Hom chom 16575  compcco 16576  SetCatcsetc 17334  ExtStrCatcestrc 17371 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 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-inf2 9095  ax-cnex 10589  ax-resscn 10590  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-addrcl 10594  ax-mulcl 10595  ax-mulrcl 10596  ax-mulcom 10597  ax-addass 10598  ax-mulass 10599  ax-distr 10600  ax-i2m1 10601  ax-1ne0 10602  ax-1rid 10603  ax-rnegex 10604  ax-rrecex 10605  ax-cnre 10606  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609  ax-pre-mulgt0 10610 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-rmo 3114  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 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-omul 8097  df-er 8279  df-ec 8281  df-qs 8285  df-map 8398  df-pm 8399  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-wun 10120  df-ni 10290  df-pli 10291  df-mi 10292  df-lti 10293  df-plpq 10326  df-mpq 10327  df-ltpq 10328  df-enq 10329  df-nq 10330  df-erq 10331  df-plq 10332  df-mq 10333  df-1nq 10334  df-rq 10335  df-ltnq 10336  df-np 10399  df-plp 10401  df-ltp 10403  df-enr 10473  df-nr 10474  df-c 10539  df-pnf 10673  df-mnf 10674  df-xr 10675  df-ltxr 10676  df-le 10677  df-sub 10868  df-neg 10869  df-nn 11633  df-2 11695  df-3 11696  df-4 11697  df-5 11698  df-6 11699  df-7 11700  df-8 11701  df-9 11702  df-n0 11893  df-z 11977  df-dec 12094  df-uz 12239  df-fz 12893  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-hom 16588  df-cco 16589  df-setc 17335  df-estrc 17372 This theorem is referenced by:  funcsetcestrc  17413
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