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Theorem catidex 17719
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 7449 . . 3 (𝑥 = 𝑋 → (𝑥𝐻𝑥) = (𝑋𝐻𝑋))
3 oveq2 7439 . . . . . 6 (𝑥 = 𝑋 → (𝑦𝐻𝑥) = (𝑦𝐻𝑋))
4 opeq2 4879 . . . . . . . . 9 (𝑥 = 𝑋 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑋⟩)
54, 1oveq12d 7449 . . . . . . . 8 (𝑥 = 𝑋 → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑋· 𝑋))
65oveqd 7448 . . . . . . 7 (𝑥 = 𝑋 → (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓))
76eqeq1d 2737 . . . . . 6 (𝑥 = 𝑋 → ((𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
83, 7raleqbidv 3344 . . . . 5 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓))
9 oveq1 7438 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
101, 1opeq12d 4886 . . . . . . . . 9 (𝑥 = 𝑋 → ⟨𝑥, 𝑥⟩ = ⟨𝑋, 𝑋⟩)
1110oveq1d 7446 . . . . . . . 8 (𝑥 = 𝑋 → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑋, 𝑋· 𝑦))
1211oveqd 7448 . . . . . . 7 (𝑥 = 𝑋 → (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔))
1312eqeq1d 2737 . . . . . 6 (𝑥 = 𝑋 → ((𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
149, 13raleqbidv 3344 . . . . 5 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
158, 14anbi12d 632 . . . 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 3176 . . 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 3345 . 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 17717 . . . 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 267 . . 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 482 . . . 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 3081 . . 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 3623 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 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cop 4637  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cat 17713
This theorem is referenced by:  catideu  17720
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