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Theorem funcsetcestrclem9 16626
 Description: Lemma 9 for funcsetcestrc 16627. (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 480 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑈 ∈ WUni)
4 eqid 2610 . . . . . 6 (Hom ‘𝑆) = (Hom ‘𝑆)
51, 2setcbas 16551 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝑆))
6 funcsetcestrc.c . . . . . . . . . . 11 𝐶 = (Base‘𝑆)
75, 6syl6reqr 2663 . . . . . . . . . 10 (𝜑𝐶 = 𝑈)
87eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑋𝐶𝑋𝑈))
98biimpcd 238 . . . . . . . 8 (𝑋𝐶 → (𝜑𝑋𝑈))
1093ad2ant1 1075 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑋𝑈))
1110impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑋𝑈)
127eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑌𝐶𝑌𝑈))
1312biimpcd 238 . . . . . . . 8 (𝑌𝐶 → (𝜑𝑌𝑈))
14133ad2ant2 1076 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑌𝑈))
1514impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑌𝑈)
161, 3, 4, 11, 15setchom 16553 . . . . 5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑋(Hom ‘𝑆)𝑌) = (𝑌𝑚 𝑋))
1716eleq2d 2673 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ↔ 𝐻 ∈ (𝑌𝑚 𝑋)))
187eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑍𝐶𝑍𝑈))
1918biimpcd 238 . . . . . . . 8 (𝑍𝐶 → (𝜑𝑍𝑈))
20193ad2ant3 1077 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑍𝑈))
2120impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑍𝑈)
221, 3, 4, 15, 21setchom 16553 . . . . 5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑌(Hom ‘𝑆)𝑍) = (𝑍𝑚 𝑌))
2322eleq2d 2673 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍) ↔ 𝐾 ∈ (𝑍𝑚 𝑌)))
2417, 23anbi12d 743 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) ↔ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))))
25 elmapi 7765 . . . . . . . . 9 (𝐾 ∈ (𝑍𝑚 𝑌) → 𝐾:𝑌𝑍)
26 elmapi 7765 . . . . . . . . 9 (𝐻 ∈ (𝑌𝑚 𝑋) → 𝐻:𝑋𝑌)
27 fco 5971 . . . . . . . . 9 ((𝐾:𝑌𝑍𝐻:𝑋𝑌) → (𝐾𝐻):𝑋𝑍)
2825, 26, 27syl2anr 494 . . . . . . . 8 ((𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌)) → (𝐾𝐻):𝑋𝑍)
2928adantl 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝐾𝐻):𝑋𝑍)
30 elmapg 7757 . . . . . . . . . 10 ((𝑍𝐶𝑋𝐶) → ((𝐾𝐻) ∈ (𝑍𝑚 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3130ancoms 468 . . . . . . . . 9 ((𝑋𝐶𝑍𝐶) → ((𝐾𝐻) ∈ (𝑍𝑚 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
32313adant2 1073 . . . . . . . 8 ((𝑋𝐶𝑌𝐶𝑍𝐶) → ((𝐾𝐻) ∈ (𝑍𝑚 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3332ad2antlr 759 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → ((𝐾𝐻) ∈ (𝑍𝑚 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3429, 33mpbird 246 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝐾𝐻) ∈ (𝑍𝑚 𝑋))
35 fvresi 6344 . . . . . 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, 6, 37, 2, 38, 39funcsetcestrclem5 16622 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑍𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍𝑚 𝑋)))
41403adantr2 1214 . . . . . . 7 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍𝑚 𝑋)))
4241adantr 480 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝑋𝐺𝑍) = ( I ↾ (𝑍𝑚 𝑋)))
433adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝑈 ∈ WUni)
44 eqid 2610 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
4511adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝑋𝑈)
4615adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝑌𝑈)
4721adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝑍𝑈)
4826ad2antrl 760 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝐻:𝑋𝑌)
4925ad2antll 761 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝐾:𝑌𝑍)
501, 43, 44, 45, 46, 47, 48, 49setcco 16556 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻) = (𝐾𝐻))
5142, 50fveq12d 6109 . . . . 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 2610 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
541, 6, 37, 2, 38funcsetcestrclem2 16618 . . . . . . . . 9 ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ 𝑈)
55543ad2antr1 1219 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑋) ∈ 𝑈)
5655adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝐹𝑋) ∈ 𝑈)
571, 6, 37, 2, 38funcsetcestrclem2 16618 . . . . . . . . 9 ((𝜑𝑌𝐶) → (𝐹𝑌) ∈ 𝑈)
58573ad2antr2 1220 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑌) ∈ 𝑈)
5958adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝐹𝑌) ∈ 𝑈)
601, 6, 37, 2, 38funcsetcestrclem2 16618 . . . . . . . . 9 ((𝜑𝑍𝐶) → (𝐹𝑍) ∈ 𝑈)
61603ad2antr3 1221 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑍) ∈ 𝑈)
6261adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝐹𝑍) ∈ 𝑈)
63 eqid 2610 . . . . . . 7 (Base‘(𝐹𝑋)) = (Base‘(𝐹𝑋))
64 eqid 2610 . . . . . . 7 (Base‘(𝐹𝑌)) = (Base‘(𝐹𝑌))
65 eqid 2610 . . . . . . 7 (Base‘(𝐹𝑍)) = (Base‘(𝐹𝑍))
66 simpll 786 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝜑)
67 3simpa 1051 . . . . . . . . . . 11 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝑋𝐶𝑌𝐶))
6867ad2antlr 759 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝑋𝐶𝑌𝐶))
69 simprl 790 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝐻 ∈ (𝑌𝑚 𝑋))
701, 6, 37, 2, 38, 39funcsetcestrclem6 16623 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌𝑚 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7166, 68, 69, 70syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
721, 6, 37funcsetcestrclem1 16617 . . . . . . . . . . . . 13 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
73723ad2antr1 1219 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
7473fveq2d 6107 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑋)) = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
75 eqid 2610 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑋⟩} = {⟨(Base‘ndx), 𝑋⟩}
76751strbas 15806 . . . . . . . . . . . . . 14 (𝑋𝐶𝑋 = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
7776eqcomd 2616 . . . . . . . . . . . . 13 (𝑋𝐶 → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
78773ad2ant1 1075 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
7978adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
8074, 79eqtrd 2644 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑋)) = 𝑋)
8180adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (Base‘(𝐹𝑋)) = 𝑋)
821, 6, 37funcsetcestrclem1 16617 . . . . . . . . . . . . 13 ((𝜑𝑌𝐶) → (𝐹𝑌) = {⟨(Base‘ndx), 𝑌⟩})
83823ad2antr2 1220 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑌) = {⟨(Base‘ndx), 𝑌⟩})
8483fveq2d 6107 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑌)) = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
85 eqid 2610 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑌⟩} = {⟨(Base‘ndx), 𝑌⟩}
86851strbas 15806 . . . . . . . . . . . . . 14 (𝑌𝐶𝑌 = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
8786eqcomd 2616 . . . . . . . . . . . . 13 (𝑌𝐶 → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
88873ad2ant2 1076 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
8988adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
9084, 89eqtrd 2644 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑌)) = 𝑌)
9190adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (Base‘(𝐹𝑌)) = 𝑌)
9271, 81, 91feq123d 5947 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹𝑋))⟶(Base‘(𝐹𝑌)) ↔ 𝐻:𝑋𝑌))
9348, 92mpbird 246 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → ((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹𝑋))⟶(Base‘(𝐹𝑌)))
94 3simpc 1053 . . . . . . . . . . 11 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝑌𝐶𝑍𝐶))
9594ad2antlr 759 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (𝑌𝐶𝑍𝐶))
96 simprr 792 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → 𝐾 ∈ (𝑍𝑚 𝑌))
971, 6, 37, 2, 38, 39funcsetcestrclem6 16623 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐶𝑍𝐶) ∧ 𝐾 ∈ (𝑍𝑚 𝑌)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9866, 95, 96, 97syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
991, 6, 37funcsetcestrclem1 16617 . . . . . . . . . . . . 13 ((𝜑𝑍𝐶) → (𝐹𝑍) = {⟨(Base‘ndx), 𝑍⟩})
100993ad2antr3 1221 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑍) = {⟨(Base‘ndx), 𝑍⟩})
101100fveq2d 6107 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑍)) = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
102 eqid 2610 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑍⟩} = {⟨(Base‘ndx), 𝑍⟩}
1031021strbas 15806 . . . . . . . . . . . . . 14 (𝑍𝐶𝑍 = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
104103eqcomd 2616 . . . . . . . . . . . . 13 (𝑍𝐶 → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
1051043ad2ant3 1077 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
106105adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
107101, 106eqtrd 2644 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑍)) = 𝑍)
108107adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (Base‘(𝐹𝑍)) = 𝑍)
10998, 91, 108feq123d 5947 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹𝑌))⟶(Base‘(𝐹𝑍)) ↔ 𝐾:𝑌𝑍))
11049, 109mpbird 246 . . . . . . 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 16593 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
11298, 71coeq12d 5208 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
113111, 112eqtrd 2644 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌𝑚 𝑋) ∧ 𝐾 ∈ (𝑍𝑚 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
11436, 51, 1133eqtr4d 2654 . . . 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 449 . . 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 229 . 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 1253 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {csn 4125  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ↑𝑚 cmap 7744  WUnicwun 9401  ndxcnx 15692  Basecbs 15695  Hom chom 15779  compcco 15780  SetCatcsetc 16548  ExtStrCatcestrc 16585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-c 9821  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-setc 16549  df-estrc 16586 This theorem is referenced by:  funcsetcestrc  16627
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