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Theorem funcestrcsetclem9 17056
Description: Lemma 9 for funcestrcsetc 17057. (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 472 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑈 ∈ WUni)
4 eqid 2765 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
51, 2estrcbas 17032 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝐸))
6 funcestrcsetc.b . . . . . . . . . . 11 𝐵 = (Base‘𝐸)
75, 6syl6reqr 2818 . . . . . . . . . 10 (𝜑𝐵 = 𝑈)
87eleq2d 2830 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
98biimpcd 240 . . . . . . . 8 (𝑋𝐵 → (𝜑𝑋𝑈))
1093ad2ant1 1163 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
1110impcom 396 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
127eleq2d 2830 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
1312biimpcd 240 . . . . . . . 8 (𝑌𝐵 → (𝜑𝑌𝑈))
14133ad2ant2 1164 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
1514impcom 396 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
16 eqid 2765 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
17 eqid 2765 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
181, 3, 4, 11, 15, 16, 17estrchom 17034 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
1918eleq2d 2830 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
207eleq2d 2830 . . . . . . . . 9 (𝜑 → (𝑍𝐵𝑍𝑈))
2120biimpcd 240 . . . . . . . 8 (𝑍𝐵 → (𝜑𝑍𝑈))
22213ad2ant3 1165 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
2322impcom 396 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
24 eqid 2765 . . . . . 6 (Base‘𝑍) = (Base‘𝑍)
251, 3, 4, 15, 23, 17, 24estrchom 17034 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
2625eleq2d 2830 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))))
2719, 26anbi12d 624 . . 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 8082 . . . . . . . . . 10 (𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29 elmapi 8082 . . . . . . . . . 10 (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30 fco 6240 . . . . . . . . . 10 ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3128, 29, 30syl2an 589 . . . . . . . . 9 ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32 fvex 6388 . . . . . . . . . 10 (Base‘𝑍) ∈ V
33 fvex 6388 . . . . . . . . . 10 (Base‘𝑋) ∈ V
3432, 33elmap 8089 . . . . . . . . 9 ((𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)) ↔ (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3531, 34sylibr 225 . . . . . . . 8 ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
3635ancoms 450 . . . . . . 7 ((𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
3736adantl 473 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
38 fvresi 6632 . . . . . 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, 6, 41, 2, 42, 43, 16, 24funcestrcsetclem5 17052 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
45443adantr2 1211 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
4645adantr 472 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
473adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑈 ∈ WUni)
48 eqid 2765 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
4911adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑋𝑈)
5015adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑌𝑈)
5123adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑍𝑈)
5229ad2antrl 719 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
5328ad2antll 720 . . . . . . 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 17037 . . . . . 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 6382 . . . . 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 2765 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
571, 40, 6, 41, 2, 42funcestrcsetclem2 17049 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
58573ad2antr1 1239 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) ∈ 𝑈)
5958adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐹𝑋) ∈ 𝑈)
601, 40, 6, 41, 2, 42funcestrcsetclem2 17049 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
61603ad2antr2 1240 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) ∈ 𝑈)
6261adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐹𝑌) ∈ 𝑈)
631, 40, 6, 41, 2, 42funcestrcsetclem2 17049 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
64633ad2antr3 1241 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) ∈ 𝑈)
6564adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐹𝑍) ∈ 𝑈)
661, 40, 6, 41, 2, 42funcestrcsetclem1 17048 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
67663ad2antr1 1239 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
681, 40, 6, 41, 2, 42funcestrcsetclem1 17048 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
69683ad2antr2 1240 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
7067, 69feq23d 6218 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7170adantr 472 . . . . . . . . 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 248 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
73 simpll 783 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝜑)
74 3simpa 1178 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7574ad2antlr 718 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑋𝐵𝑌𝐵))
76 simprl 787 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
771, 40, 6, 41, 2, 42, 43, 16, 17funcestrcsetclem6 17053 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7873, 75, 76, 77syl3anc 1490 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7978feq1d 6208 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
8072, 79mpbird 248 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌))
811, 40, 6, 41, 2, 42funcestrcsetclem1 17048 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
82813ad2antr3 1241 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8369, 82feq23d 6218 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8483adantr 472 . . . . . . . . 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 248 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
86 3simpc 1182 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8786ad2antlr 718 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑌𝐵𝑍𝐵))
88 simprr 789 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
891, 40, 6, 41, 2, 42, 43, 17, 24funcestrcsetclem6 17053 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9073, 87, 88, 89syl3anc 1490 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9190feq1d 6208 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9285, 91mpbird 248 . . . . . . 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 17000 . . . . . 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 5455 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9593, 94eqtrd 2799 . . . . 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 2809 . . . 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 401 . . 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 231 . 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 1145 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cop 4340  cmpt 4888   I cid 5184  cres 5279  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑚 cmap 8060  WUnicwun 9775  Basecbs 16132  Hom chom 16227  compcco 16228  SetCatcsetc 16992  ExtStrCatcestrc 17029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-wun 9777  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-hom 16240  df-cco 16241  df-setc 16993  df-estrc 17030
This theorem is referenced by:  funcestrcsetc  17057
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