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Theorem catideu 16264
Description: Each object in a category has a unique 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
catideu (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 catideu
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 catidex.b . . 3 𝐵 = (Base‘𝐶)
2 catidex.h . . 3 𝐻 = (Hom ‘𝐶)
3 catidex.o . . 3 · = (comp‘𝐶)
4 catidex.c . . 3 (𝜑𝐶 ∈ Cat)
5 catidex.x . . 3 (𝜑𝑋𝐵)
61, 2, 3, 4, 5catidex 16263 . 2 (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
7 oveq1 6617 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦𝐻𝑋) = (𝑋𝐻𝑋))
8 opeq1 4375 . . . . . . . . . . 11 (𝑦 = 𝑋 → ⟨𝑦, 𝑋⟩ = ⟨𝑋, 𝑋⟩)
98oveq1d 6625 . . . . . . . . . 10 (𝑦 = 𝑋 → (⟨𝑦, 𝑋· 𝑋) = (⟨𝑋, 𝑋· 𝑋))
109oveqd 6627 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓))
1110eqeq1d 2623 . . . . . . . 8 (𝑦 = 𝑋 → ((𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
127, 11raleqbidv 3144 . . . . . . 7 (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
13 oveq2 6618 . . . . . . . 8 (𝑦 = 𝑋 → (𝑋𝐻𝑦) = (𝑋𝐻𝑋))
14 oveq2 6618 . . . . . . . . . 10 (𝑦 = 𝑋 → (⟨𝑋, 𝑋· 𝑦) = (⟨𝑋, 𝑋· 𝑋))
1514oveqd 6627 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑋)𝑔))
1615eqeq1d 2623 . . . . . . . 8 (𝑦 = 𝑋 → ((𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓))
1713, 16raleqbidv 3144 . . . . . . 7 (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓))
1812, 17anbi12d 746 . . . . . 6 (𝑦 = 𝑋 → ((∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓)))
1918rspcv 3294 . . . . 5 (𝑋𝐵 → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓)))
205, 19syl 17 . . . 4 (𝜑 → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓)))
2120ralrimivw 2962 . . 3 (𝜑 → ∀𝑔 ∈ (𝑋𝐻𝑋)(∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓)))
22 an3 867 . . . . . . 7 (((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓)) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓))
23 oveq2 6618 . . . . . . . . . 10 (𝑓 = → (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋· 𝑋)))
24 id 22 . . . . . . . . . 10 (𝑓 = 𝑓 = )
2523, 24eqeq12d 2636 . . . . . . . . 9 (𝑓 = → ((𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋· 𝑋)) = ))
2625rspcv 3294 . . . . . . . 8 ( ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 → (𝑔(⟨𝑋, 𝑋· 𝑋)) = ))
27 oveq1 6617 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓(⟨𝑋, 𝑋· 𝑋)) = (𝑔(⟨𝑋, 𝑋· 𝑋)))
28 id 22 . . . . . . . . . 10 (𝑓 = 𝑔𝑓 = 𝑔)
2927, 28eqeq12d 2636 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔))
3029rspcv 3294 . . . . . . . 8 (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓 → (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔))
3126, 30im2anan9r 880 . . . . . . 7 ((𝑔 ∈ (𝑋𝐻𝑋) ∧ ∈ (𝑋𝐻𝑋)) → ((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓) → ((𝑔(⟨𝑋, 𝑋· 𝑋)) = ∧ (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔)))
32 eqtr2 2641 . . . . . . . 8 (((𝑔(⟨𝑋, 𝑋· 𝑋)) = ∧ (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔) → = 𝑔)
3332equcomd 1943 . . . . . . 7 (((𝑔(⟨𝑋, 𝑋· 𝑋)) = ∧ (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔) → 𝑔 = )
3422, 31, 33syl56 36 . . . . . 6 ((𝑔 ∈ (𝑋𝐻𝑋) ∧ ∈ (𝑋𝐻𝑋)) → (((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓)) → 𝑔 = ))
3534rgen2a 2972 . . . . 5 𝑔 ∈ (𝑋𝐻𝑋)∀ ∈ (𝑋𝐻𝑋)(((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓)) → 𝑔 = )
3635a1i 11 . . . 4 (𝜑 → ∀𝑔 ∈ (𝑋𝐻𝑋)∀ ∈ (𝑋𝐻𝑋)(((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓)) → 𝑔 = ))
37 oveq1 6617 . . . . . . . 8 (𝑔 = → (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = ((⟨𝑋, 𝑋· 𝑋)𝑓))
3837eqeq1d 2623 . . . . . . 7 (𝑔 = → ((𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ↔ ((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
3938ralbidv 2981 . . . . . 6 (𝑔 = → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
40 oveq2 6618 . . . . . . . 8 (𝑔 = → (𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑋)))
4140eqeq1d 2623 . . . . . . 7 (𝑔 = → ((𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓))
4241ralbidv 2981 . . . . . 6 (𝑔 = → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓))
4339, 42anbi12d 746 . . . . 5 (𝑔 = → ((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓)))
4443rmo4 3385 . . . 4 (∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ↔ ∀𝑔 ∈ (𝑋𝐻𝑋)∀ ∈ (𝑋𝐻𝑋)(((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓)) → 𝑔 = ))
4536, 44sylibr 224 . . 3 (𝜑 → ∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓))
46 rmoim 3393 . . 3 (∀𝑔 ∈ (𝑋𝐻𝑋)(∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓)) → (∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓) → ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
4721, 45, 46sylc 65 . 2 (𝜑 → ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
48 reu5 3151 . 2 (∃!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) ∧ ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))
496, 47, 48sylanbrc 697 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  ∃*wrmo 2910  cop 4159  cfv 5852  (class class class)co 6610  Basecbs 15788  Hom chom 15880  compcco 15881  Catccat 16253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-cat 16257
This theorem is referenced by:  catidd  16269  catidcl  16271  catlid  16272  catrid  16273
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