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Theorem funcestrcsetclem9 16981
Description: Lemma 9 for funcestrcsetc 16982. (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 2752 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
51, 2estrcbas 16958 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝐸))
6 funcestrcsetc.b . . . . . . . . . . 11 𝐵 = (Base‘𝐸)
75, 6syl6reqr 2805 . . . . . . . . . 10 (𝜑𝐵 = 𝑈)
87eleq2d 2817 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
98biimpcd 239 . . . . . . . 8 (𝑋𝐵 → (𝜑𝑋𝑈))
1093ad2ant1 1127 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
1110impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
127eleq2d 2817 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
1312biimpcd 239 . . . . . . . 8 (𝑌𝐵 → (𝜑𝑌𝑈))
14133ad2ant2 1128 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
1514impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
16 eqid 2752 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
17 eqid 2752 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
181, 3, 4, 11, 15, 16, 17estrchom 16960 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
1918eleq2d 2817 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
207eleq2d 2817 . . . . . . . . 9 (𝜑 → (𝑍𝐵𝑍𝑈))
2120biimpcd 239 . . . . . . . 8 (𝑍𝐵 → (𝜑𝑍𝑈))
22213ad2ant3 1129 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
2322impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
24 eqid 2752 . . . . . 6 (Base‘𝑍) = (Base‘𝑍)
251, 3, 4, 15, 23, 17, 24estrchom 16960 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
2625eleq2d 2817 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))))
2719, 26anbi12d 749 . . 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 8037 . . . . . . . . . 10 (𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29 elmapi 8037 . . . . . . . . . 10 (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30 fco 6211 . . . . . . . . . 10 ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3128, 29, 30syl2an 495 . . . . . . . . 9 ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32 fvex 6354 . . . . . . . . . 10 (Base‘𝑍) ∈ V
33 fvex 6354 . . . . . . . . . 10 (Base‘𝑋) ∈ V
3432, 33elmap 8044 . . . . . . . . 9 ((𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)) ↔ (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3531, 34sylibr 224 . . . . . . . 8 ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
3635ancoms 468 . . . . . . 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 6595 . . . . . 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 16977 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
45443adantr2 1173 . . . . . . 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 2752 . . . . . . 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 766 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
5328ad2antll 767 . . . . . . 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 16963 . . . . . 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 6350 . . . . 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 2752 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
571, 40, 6, 41, 2, 42funcestrcsetclem2 16974 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
58573ad2antr1 1201 . . . . . . . 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 16974 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
61603ad2antr2 1202 . . . . . . . 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 16974 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
64633ad2antr3 1203 . . . . . . . 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 16973 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
67663ad2antr1 1201 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
681, 40, 6, 41, 2, 42funcestrcsetclem1 16973 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
69683ad2antr2 1202 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
7067, 69feq23d 6193 . . . . . . . . . 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 247 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
73 simpll 807 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝜑)
74 3simpa 1142 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7574ad2antlr 765 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑋𝐵𝑌𝐵))
76 simprl 811 . . . . . . . . . 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 16978 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7873, 75, 76, 77syl3anc 1473 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7978feq1d 6183 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
8072, 79mpbird 247 . . . . . . 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 16973 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
82813ad2antr3 1203 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8369, 82feq23d 6193 . . . . . . . . . 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 247 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
86 3simpc 1146 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8786ad2antlr 765 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑌𝐵𝑍𝐵))
88 simprr 813 . . . . . . . . . 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 16978 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9073, 87, 88, 89syl3anc 1473 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9190feq1d 6183 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9285, 91mpbird 247 . . . . . . 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 16926 . . . . . 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 5434 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9593, 94eqtrd 2786 . . . . 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 2796 . . . 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 449 . . 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 230 . 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 1109 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  cop 4319  cmpt 4873   I cid 5165  cres 5260  ccom 5262  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  𝑚 cmap 8015  WUnicwun 9706  Basecbs 16051  Hom chom 16146  compcco 16147  SetCatcsetc 16918  ExtStrCatcestrc 16955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-wun 9708  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-hom 16160  df-cco 16161  df-setc 16919  df-estrc 16956
This theorem is referenced by:  funcestrcsetc  16982
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