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Theorem funcestrcsetclem9 17392
 Description: Lemma 9 for funcestrcsetc 17393. (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 484 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑈 ∈ WUni)
4 eqid 2798 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
5 funcestrcsetc.b . . . . . . . . . . 11 𝐵 = (Base‘𝐸)
61, 2estrcbas 17369 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝐸))
75, 6eqtr4id 2852 . . . . . . . . . 10 (𝜑𝐵 = 𝑈)
87eleq2d 2875 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
98biimpcd 252 . . . . . . . 8 (𝑋𝐵 → (𝜑𝑋𝑈))
1093ad2ant1 1130 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
1110impcom 411 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
127eleq2d 2875 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
1312biimpcd 252 . . . . . . . 8 (𝑌𝐵 → (𝜑𝑌𝑈))
14133ad2ant2 1131 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
1514impcom 411 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
16 eqid 2798 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
17 eqid 2798 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
181, 3, 4, 11, 15, 16, 17estrchom 17371 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
1918eleq2d 2875 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
207eleq2d 2875 . . . . . . . . 9 (𝜑 → (𝑍𝐵𝑍𝑈))
2120biimpcd 252 . . . . . . . 8 (𝑍𝐵 → (𝜑𝑍𝑈))
22213ad2ant3 1132 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
2322impcom 411 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
24 eqid 2798 . . . . . 6 (Base‘𝑍) = (Base‘𝑍)
251, 3, 4, 15, 23, 17, 24estrchom 17371 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑m (Base‘𝑌)))
2625eleq2d 2875 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))))
2719, 26anbi12d 633 . . 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 8413 . . . . . . . . . 10 (𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29 elmapi 8413 . . . . . . . . . 10 (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30 fco 6505 . . . . . . . . . 10 ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3128, 29, 30syl2an 598 . . . . . . . . 9 ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32 fvex 6658 . . . . . . . . . 10 (Base‘𝑍) ∈ V
33 fvex 6658 . . . . . . . . . 10 (Base‘𝑋) ∈ V
3432, 33elmap 8420 . . . . . . . . 9 ((𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)) ↔ (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3531, 34sylibr 237 . . . . . . . 8 ((𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
3635ancoms 462 . . . . . . 7 ((𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
3736adantl 485 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑m (Base‘𝑋)))
38 fvresi 6912 . . . . . 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 17388 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
45443adantr2 1167 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
4645adantr 484 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑m (Base‘𝑋))))
473adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑈 ∈ WUni)
48 eqid 2798 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
4911adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑋𝑈)
5015adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑌𝑈)
5123adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝑍𝑈)
5229ad2antrl 727 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
5328ad2antll 728 . . . . . . 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 17374 . . . . . 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 6652 . . . . 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 2798 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
571, 40, 5, 41, 2, 42funcestrcsetclem2 17385 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
58573ad2antr1 1185 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) ∈ 𝑈)
5958adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹𝑋) ∈ 𝑈)
601, 40, 5, 41, 2, 42funcestrcsetclem2 17385 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
61603ad2antr2 1186 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) ∈ 𝑈)
6261adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹𝑌) ∈ 𝑈)
631, 40, 5, 41, 2, 42funcestrcsetclem2 17385 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
64633ad2antr3 1187 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) ∈ 𝑈)
6564adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝐹𝑍) ∈ 𝑈)
661, 40, 5, 41, 2, 42funcestrcsetclem1 17384 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
67663ad2antr1 1185 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
681, 40, 5, 41, 2, 42funcestrcsetclem1 17384 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
69683ad2antr2 1186 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
7067, 69feq23d 6482 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7170adantr 484 . . . . . . . . 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 260 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
73 simpll 766 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝜑)
74 3simpa 1145 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7574ad2antlr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑋𝐵𝑌𝐵))
76 simprl 770 . . . . . . . . . 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 17389 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7873, 75, 76, 77syl3anc 1368 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7978feq1d 6472 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
8072, 79mpbird 260 . . . . . . 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 17384 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
82813ad2antr3 1187 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8369, 82feq23d 6482 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8483adantr 484 . . . . . . . . 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 260 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
86 3simpc 1147 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8786ad2antlr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (𝑌𝐵𝑍𝐵))
88 simprr 772 . . . . . . . . . 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 17389 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9073, 87, 88, 89syl3anc 1368 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9190feq1d 6472 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9285, 91mpbird 260 . . . . . . 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 17337 . . . . . 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 5699 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9593, 94eqtrd 2833 . . . . 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 2843 . . . 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 416 . . 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 243 . 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 1114 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   ↦ cmpt 5110   I cid 5424   ↾ cres 5521   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑m cmap 8391  WUnicwun 10113  Basecbs 16477  Hom chom 16570  compcco 16571  SetCatcsetc 17329  ExtStrCatcestrc 17366 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 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605 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-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 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-wun 10115  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-nn 11628  df-2 11690  df-3 11691  df-4 11692  df-5 11693  df-6 11694  df-7 11695  df-8 11696  df-9 11697  df-n0 11888  df-z 11972  df-dec 12089  df-uz 12234  df-fz 12888  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-hom 16583  df-cco 16584  df-setc 17330  df-estrc 17367 This theorem is referenced by:  funcestrcsetc  17393
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