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Theorem catidex 16940
 Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catidex.b 𝐵 = (Base‘𝐶)
catidex.h 𝐻 = (Hom ‘𝐶)
catidex.o · = (comp‘𝐶)
catidex.c (𝜑𝐶 ∈ Cat)
catidex.x (𝜑𝑋𝐵)
Assertion
Ref Expression
catidex (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
Distinct variable groups:   𝑓,𝑔,𝑦,𝐵   𝐶,𝑓,𝑔,𝑦   𝜑,𝑔   𝑓,𝑋,𝑔,𝑦   𝑓,𝐻,𝑔,𝑦   · ,𝑓,𝑔,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑓)

Proof of Theorem catidex
Dummy variables 𝑘 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
21, 1oveq12d 7157 . . 3 (𝑥 = 𝑋 → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
3 oveq2 7147 . . . . . 6 (𝑥 = 𝑋 → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
4 opeq2 4768 . . . . . . . . 9 (𝑥 = 𝑋 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
54, 1oveq12d 7157 . . . . . . . 8 (𝑥 = 𝑋 → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑋· 𝑋))
65oveqd 7156 . . . . . . 7 (𝑥 = 𝑋 → (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓))
76eqeq1d 2803 . . . . . 6 (𝑥 = 𝑋 → ((𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
83, 7raleqbidv 3357 . . . . 5 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
9 oveq1 7146 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
101, 1opeq12d 4776 . . . . . . . . 9 (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
1110oveq1d 7154 . . . . . . . 8 (𝑥 = 𝑋 → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑋, 𝑋· 𝑦))
1211oveqd 7156 . . . . . . 7 (𝑥 = 𝑋 → (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔))
1312eqeq1d 2803 . . . . . 6 (𝑥 = 𝑋 → ((𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
149, 13raleqbidv 3357 . . . . 5 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
158, 14anbi12d 633 . . . 4 (𝑥 = 𝑋 → ((∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
1615ralbidv 3165 . . 3 (𝑥 = 𝑋 → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
172, 16rexeqbidv 3358 . 2 (𝑥 = 𝑋 → (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
18 catidex.c . . 3 (𝜑𝐶 ∈ Cat)
19 catidex.b . . . . 5 𝐵 = (Base‘𝐶)
20 catidex.h . . . . 5 𝐻 = (Hom ‘𝐶)
21 catidex.o . . . . 5 · = (comp‘𝐶)
2219, 20, 21iscat 16938 . . . 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) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))
2322ibi 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) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
24 simpl 486 . . . 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) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) → ∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
2524ralimi 3131 . . 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) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) → ∀𝑥𝐵𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
2618, 23, 253syl 18 . 2 (𝜑 → ∀𝑥𝐵𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
27 catidex.x . 2 (𝜑𝑋𝐵)
2817, 26, 27rspcdva 3576 1 (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110  ⟨cop 4534  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  Hom chom 16571  compcco 16572  Catccat 16930 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-cat 16934 This theorem is referenced by:  catideu  16941
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