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Theorem catcocl 16947
Description: Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catcocl.b 𝐵 = (Base‘𝐶)
catcocl.h 𝐻 = (Hom ‘𝐶)
catcocl.o · = (comp‘𝐶)
catcocl.c (𝜑𝐶 ∈ Cat)
catcocl.x (𝜑𝑋𝐵)
catcocl.y (𝜑𝑌𝐵)
catcocl.z (𝜑𝑍𝐵)
catcocl.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
catcocl.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
catcocl (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍))

Proof of Theorem catcocl
Dummy variables 𝑓 𝑔 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcocl.c . . 3 (𝜑𝐶 ∈ Cat)
2 catcocl.b . . . . 5 𝐵 = (Base‘𝐶)
3 catcocl.h . . . . 5 𝐻 = (Hom ‘𝐶)
4 catcocl.o . . . . 5 · = (comp‘𝐶)
52, 3, 4iscat 16934 . . . 4 (𝐶 ∈ Cat → (𝐶 ∈ Cat ↔ ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))
65ibi 270 . . 3 (𝐶 ∈ Cat → ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
7 simpl 486 . . . . . . 7 (((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
872ralimi 3153 . . . . . 6 (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
982ralimi 3153 . . . . 5 (∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
109adantl 485 . . . 4 ((∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
1110ralimi 3152 . . 3 (∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
121, 6, 113syl 18 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
13 catcocl.x . . 3 (𝜑𝑋𝐵)
14 catcocl.y . . . . 5 (𝜑𝑌𝐵)
1514adantr 484 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
16 catcocl.z . . . . . 6 (𝜑𝑍𝐵)
1716ad2antrr 725 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍𝐵)
18 catcocl.f . . . . . . . 8 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
1918ad3antrrr 729 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐻𝑌))
20 simpllr 775 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
21 simplr 768 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
2220, 21oveq12d 7158 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2319, 22eleqtrrd 2917 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐻𝑦))
24 catcocl.g . . . . . . . . . 10 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
2524ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑌𝐻𝑍))
26 simpr 488 . . . . . . . . . 10 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
2721, 26oveq12d 7158 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍))
2825, 27eleqtrrd 2917 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑦𝐻𝑧))
2928adantr 484 . . . . . . 7 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐻𝑧))
30 simp-5r 785 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑥 = 𝑋)
31 simp-4r 783 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑌)
3230, 31opeq12d 4786 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
33 simpllr 775 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑧 = 𝑍)
3432, 33oveq12d 7158 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑋, 𝑌· 𝑍))
35 simpr 488 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
36 simplr 768 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
3734, 35, 36oveq123d 7161 . . . . . . . 8 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
3830, 33oveq12d 7158 . . . . . . . 8 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐻𝑧) = (𝑋𝐻𝑍))
3937, 38eleq12d 2908 . . . . . . 7 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ↔ (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4029, 39rspcdv 3590 . . . . . 6 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4123, 40rspcimdv 3588 . . . . 5 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4217, 41rspcimdv 3588 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4315, 42rspcimdv 3588 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4413, 43rspcimdv 3588 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4512, 44mpd 15 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  cop 4545  cfv 6334  (class class class)co 7140  Basecbs 16474  Hom chom 16567  compcco 16568  Catccat 16926
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-cat 16930
This theorem is referenced by:  oppccatid  16980  ismon2  16995  isepi2  17002  sectco  17017  monsect  17044  catsubcat  17100  issubc3  17110  fullsubc  17111  idfucl  17142  cofucl  17149  fthsect  17186  fthmon  17188  fuccocl  17225  invfuc  17235  2initoinv  17261  initoeu2lem0  17264  initoeu2lem1  17265  initoeu2  17267  2termoinv  17268  coahom  17321  catcisolem  17357  xpccatid  17429  1stfcl  17438  2ndfcl  17439  prfcl  17444  evlfcllem  17462  evlfcl  17463  curf1cl  17469  curfcl  17473  hofcllem  17499  hofcl  17500  yon12  17506  hofpropd  17508  yonedalem4c  17518  bj-endmnd  34693  srhmsubc  44640  srhmsubcALTV  44658
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