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Theorem funcestrcsetclem9 18036
Description: Lemma 9 for funcestrcsetc 18037. (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 (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
Assertion
Ref Expression
funcestrcsetclem9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((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
StepHypRef Expression
1 funcestrcsetc.e . . . . . 6 𝐸 = (ExtStrCat‘𝑈)
2 funcestrcsetc.u . . . . . . 7 (𝜑𝑈 ∈ WUni)
32adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑈 ∈ WUni)
4 eqid 2736 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
5 funcestrcsetc.b . . . . . . . . . . 11 𝐵 = (Base‘𝐸)
61, 2estrcbas 18012 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝐸))
75, 6eqtr4id 2795 . . . . . . . . . 10 (𝜑𝐵 = 𝑈)
87eleq2d 2823 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
98biimpcd 248 . . . . . . . 8 (𝑋𝐵 → (𝜑𝑋𝑈))
1093ad2ant1 1133 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
1110impcom 408 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
127eleq2d 2823 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
1312biimpcd 248 . . . . . . . 8 (𝑌𝐵 → (𝜑𝑌𝑈))
14133ad2ant2 1134 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
1514impcom 408 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
16 eqid 2736 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
17 eqid 2736 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
181, 3, 4, 11, 15, 16, 17estrchom 18014 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
1918eleq2d 2823 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
207eleq2d 2823 . . . . . . . . 9 (𝜑 → (𝑍𝐵𝑍𝑈))
2120biimpcd 248 . . . . . . . 8 (𝑍𝐵 → (𝜑𝑍𝑈))
22213ad2ant3 1135 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
2322impcom 408 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
24 eqid 2736 . . . . . 6 (Base‘𝑍) = (Base‘𝑍)
251, 3, 4, 15, 23, 17, 24estrchom 18014 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑m (Base‘𝑌)))
2625eleq2d 2823 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))))
2719, 26anbi12d 631 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍)) ↔ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))))
28 elmapi 8787 . . . . . . . . . 10 (𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29 elmapi 8787 . . . . . . . . . 10 (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30 fco 6692 . . . . . . . . . 10 ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3128, 29, 30syl2an 596 . . . . . . . . 9 ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32 fvex 6855 . . . . . . . . . 10 (Base‘𝑍) ∈ V
33 fvex 6855 . . . . . . . . . 10 (Base‘𝑋) ∈ V
3432, 33elmap 8809 . . . . . . . . 9 ((𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) ↔ (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3531, 34sylibr 233 . . . . . . . 8 ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
3635ancoms 459 . . . . . . 7 ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
3736adantl 482 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
38 fvresi 7119 . . . . . 6 ((𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) → (( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋)))‘(𝐾𝐻)) = (𝐾𝐻))
3937, 38syl 17 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋)))‘(𝐾𝐻)) = (𝐾𝐻))
40 funcestrcsetc.s . . . . . . . . 9 𝑆 = (SetCat‘𝑈)
41 funcestrcsetc.c . . . . . . . . 9 𝐶 = (Base‘𝑆)
42 funcestrcsetc.f . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
43 funcestrcsetc.g . . . . . . . . 9 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
441, 40, 5, 41, 2, 42, 43, 16, 24funcestrcsetclem5 18032 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
45443adantr2 1170 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
4645adantr 481 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
473adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑈 ∈ WUni)
48 eqid 2736 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
4911adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑋𝑈)
5015adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑌𝑈)
5123adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑍𝑈)
5229ad2antrl 726 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
5328ad2antll 727 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
541, 47, 48, 49, 50, 51, 16, 17, 24, 52, 53estrcco 18017 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻) = (𝐾𝐻))
5546, 54fveq12d 6849 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (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)))‘(𝐾𝐻)))
56 eqid 2736 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
571, 40, 5, 41, 2, 42funcestrcsetclem2 18029 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
58573ad2antr1 1188 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) ∈ 𝑈)
5958adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹𝑋) ∈ 𝑈)
601, 40, 5, 41, 2, 42funcestrcsetclem2 18029 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
61603ad2antr2 1189 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) ∈ 𝑈)
6261adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹𝑌) ∈ 𝑈)
631, 40, 5, 41, 2, 42funcestrcsetclem2 18029 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
64633ad2antr3 1190 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) ∈ 𝑈)
6564adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹𝑍) ∈ 𝑈)
661, 40, 5, 41, 2, 42funcestrcsetclem1 18028 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
67663ad2antr1 1188 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
681, 40, 5, 41, 2, 42funcestrcsetclem1 18028 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
69683ad2antr2 1189 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
7067, 69feq23d 6663 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7170adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7252, 71mpbird 256 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
73 simpll 765 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝜑)
74 3simpa 1148 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7574ad2antlr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑋𝐵𝑌𝐵))
76 simprl 769 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
771, 40, 5, 41, 2, 42, 43, 16, 17funcestrcsetclem6 18033 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7873, 75, 76, 77syl3anc 1371 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7978feq1d 6653 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
8072, 79mpbird 256 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌))
811, 40, 5, 41, 2, 42funcestrcsetclem1 18028 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
82813ad2antr3 1190 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8369, 82feq23d 6663 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8483adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8553, 84mpbird 256 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
86 3simpc 1150 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8786ad2antlr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑌𝐵𝑍𝐵))
88 simprr 771 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))
891, 40, 5, 41, 2, 42, 43, 17, 24funcestrcsetclem6 18033 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9073, 87, 88, 89syl3anc 1371 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9190feq1d 6653 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9285, 91mpbird 256 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍))
9340, 47, 56, 59, 62, 65, 80, 92setcco 17969 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
9490, 78coeq12d 5820 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9593, 94eqtrd 2776 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9639, 55, 953eqtr4d 2786 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
9796ex 413 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
9827, 97sylbid 239 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
99983impia 1117 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cop 4592  cmpt 5188   I cid 5530  cres 5635  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  WUnicwun 10636  Basecbs 17083  Hom chom 17144  compcco 17145  SetCatcsetc 17961  ExtStrCatcestrc 18009
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wun 10638  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-setc 17962  df-estrc 18010
This theorem is referenced by:  funcestrcsetc  18037
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