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Theorem catideu 16458
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 16457 . 2 (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))
7 oveq1 6772 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦𝐻𝑋) = (𝑋𝐻𝑋))
8 opeq1 4509 . . . . . . . . . . 11 (𝑦 = 𝑋 → ⟨𝑦, 𝑋⟩ = ⟨𝑋, 𝑋⟩)
98oveq1d 6780 . . . . . . . . . 10 (𝑦 = 𝑋 → (⟨𝑦, 𝑋· 𝑋) = (⟨𝑋, 𝑋· 𝑋))
109oveqd 6782 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓))
1110eqeq1d 2726 . . . . . . . 8 (𝑦 = 𝑋 → ((𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
127, 11raleqbidv 3255 . . . . . . 7 (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
13 oveq2 6773 . . . . . . . 8 (𝑦 = 𝑋 → (𝑋𝐻𝑦) = (𝑋𝐻𝑋))
14 oveq2 6773 . . . . . . . . . 10 (𝑦 = 𝑋 → (⟨𝑋, 𝑋· 𝑦) = (⟨𝑋, 𝑋· 𝑋))
1514oveqd 6782 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑋)𝑔))
1615eqeq1d 2726 . . . . . . . 8 (𝑦 = 𝑋 → ((𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓))
1713, 16raleqbidv 3255 . . . . . . 7 (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓))
1812, 17anbi12d 749 . . . . . 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 3409 . . . . 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 3069 . . 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 903 . . . . . . 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 6773 . . . . . . . . . 10 (𝑓 = → (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋· 𝑋)))
24 id 22 . . . . . . . . . 10 (𝑓 = 𝑓 = )
2523, 24eqeq12d 2739 . . . . . . . . 9 (𝑓 = → ((𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋· 𝑋)) = ))
2625rspcv 3409 . . . . . . . 8 ( ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 → (𝑔(⟨𝑋, 𝑋· 𝑋)) = ))
27 oveq1 6772 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓(⟨𝑋, 𝑋· 𝑋)) = (𝑔(⟨𝑋, 𝑋· 𝑋)))
28 id 22 . . . . . . . . . 10 (𝑓 = 𝑔𝑓 = 𝑔)
2927, 28eqeq12d 2739 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔))
3029rspcv 3409 . . . . . . . 8 (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓 → (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔))
3126, 30im2anan9r 917 . . . . . . 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 2744 . . . . . . . 8 (((𝑔(⟨𝑋, 𝑋· 𝑋)) = ∧ (𝑔(⟨𝑋, 𝑋· 𝑋)) = 𝑔) → = 𝑔)
3332equcomd 2065 . . . . . . 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 3079 . . . . 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 6772 . . . . . . . 8 (𝑔 = → (𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = ((⟨𝑋, 𝑋· 𝑋)𝑓))
3837eqeq1d 2726 . . . . . . 7 (𝑔 = → ((𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ↔ ((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
3938ralbidv 3088 . . . . . 6 (𝑔 = → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)((⟨𝑋, 𝑋· 𝑋)𝑓) = 𝑓))
40 oveq2 6773 . . . . . . . 8 (𝑔 = → (𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = (𝑓(⟨𝑋, 𝑋· 𝑋)))
4140eqeq1d 2726 . . . . . . 7 (𝑔 = → ((𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓))
4241ralbidv 3088 . . . . . 6 (𝑔 = → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋· 𝑋)) = 𝑓))
4339, 42anbi12d 749 . . . . 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 3505 . . . 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 3513 . . 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 3262 . 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 701 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 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  ∃!wreu 3016  ∃*wrmo 3017  cop 4291  cfv 6001  (class class class)co 6765  Basecbs 15980  Hom chom 16075  compcco 16076  Catccat 16447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-nul 4897
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-iota 5964  df-fv 6009  df-ov 6768  df-cat 16451
This theorem is referenced by:  catidd  16463  catidcl  16465  catlid  16466  catrid  16467
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