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Theorem funcestrcsetclem9 16611
 Description: Lemma 9 for funcestrcsetc 16612. (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 480 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑈 ∈ WUni)
4 eqid 2610 . . . . . 6 (Hom ‘𝐸) = (Hom ‘𝐸)
51, 2estrcbas 16588 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝐸))
6 funcestrcsetc.b . . . . . . . . . . 11 𝐵 = (Base‘𝐸)
75, 6syl6reqr 2663 . . . . . . . . . 10 (𝜑𝐵 = 𝑈)
87eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
98biimpcd 238 . . . . . . . 8 (𝑋𝐵 → (𝜑𝑋𝑈))
1093ad2ant1 1075 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
1110impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
127eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
1312biimpcd 238 . . . . . . . 8 (𝑌𝐵 → (𝜑𝑌𝑈))
14133ad2ant2 1076 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
1514impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
16 eqid 2610 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
17 eqid 2610 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
181, 3, 4, 11, 15, 16, 17estrchom 16590 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
1918eleq2d 2673 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ↔ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
207eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑍𝐵𝑍𝑈))
2120biimpcd 238 . . . . . . . 8 (𝑍𝐵 → (𝜑𝑍𝑈))
22213ad2ant3 1077 . . . . . . 7 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
2322impcom 445 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
24 eqid 2610 . . . . . 6 (Base‘𝑍) = (Base‘𝑍)
251, 3, 4, 15, 23, 17, 24estrchom 16590 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝐸)𝑍) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
2625eleq2d 2673 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍) ↔ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))))
2719, 26anbi12d 743 . . 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 7765 . . . . . . . . . 10 (𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
29 elmapi 7765 . . . . . . . . . 10 (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
30 fco 5971 . . . . . . . . . 10 ((𝐾:(Base‘𝑌)⟶(Base‘𝑍) ∧ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3128, 29, 30syl2an 493 . . . . . . . . 9 ((𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
32 fvex 6113 . . . . . . . . . 10 (Base‘𝑍) ∈ V
33 fvex 6113 . . . . . . . . . 10 (Base‘𝑋) ∈ V
3432, 33elmap 7772 . . . . . . . . 9 ((𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)) ↔ (𝐾𝐻):(Base‘𝑋)⟶(Base‘𝑍))
3531, 34sylibr 223 . . . . . . . 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 481 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐾𝐻) ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑋)))
38 fvresi 6344 . . . . . 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 16607 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
45443adantr2 1214 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
4645adantr 480 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑋𝐺𝑍) = ( I ↾ ((Base‘𝑍) ↑𝑚 (Base‘𝑋))))
473adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑈 ∈ WUni)
48 eqid 2610 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
4911adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑋𝑈)
5015adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑌𝑈)
5123adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝑍𝑈)
5229ad2antrl 760 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
5328ad2antll 761 . . . . . . 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 16593 . . . . . 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 6109 . . . . 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 2610 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
571, 40, 6, 41, 2, 42funcestrcsetclem2 16604 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
58573ad2antr1 1219 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) ∈ 𝑈)
5958adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐹𝑋) ∈ 𝑈)
601, 40, 6, 41, 2, 42funcestrcsetclem2 16604 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
61603ad2antr2 1220 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) ∈ 𝑈)
6261adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐹𝑌) ∈ 𝑈)
631, 40, 6, 41, 2, 42funcestrcsetclem2 16604 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
64633ad2antr3 1221 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) ∈ 𝑈)
6564adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝐹𝑍) ∈ 𝑈)
661, 40, 6, 41, 2, 42funcestrcsetclem1 16603 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
67663ad2antr1 1219 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
681, 40, 6, 41, 2, 42funcestrcsetclem1 16603 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
69683ad2antr2 1220 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
7067, 69feq23d 5953 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7170adantr 480 . . . . . . . . 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 246 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
73 simpll 786 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝜑)
74 3simpa 1051 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7574ad2antlr 759 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑋𝐵𝑌𝐵))
76 simprl 790 . . . . . . . . . 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 16608 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7873, 75, 76, 77syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7978feq1d 5943 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
8072, 79mpbird 246 . . . . . . 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 16603 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
82813ad2antr3 1221 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8369, 82feq23d 5953 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8483adantr 480 . . . . . . . . 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 246 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
86 3simpc 1053 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8786ad2antlr 759 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (𝑌𝐵𝑍𝐵))
88 simprr 792 . . . . . . . . . 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 16608 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9073, 87, 88, 89syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9190feq1d 5943 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9285, 91mpbird 246 . . . . . . 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 16556 . . . . . 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 5208 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∧ 𝐾 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9593, 94eqtrd 2644 . . . . 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 2654 . . . 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 229 . 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 1253 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  WUnicwun 9401  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-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-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-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  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:  funcestrcsetc  16612
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