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Theorem catidd 16945
Description: Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catidd.b (𝜑𝐵 = (Base‘𝐶))
catidd.h (𝜑𝐻 = (Hom ‘𝐶))
catidd.o (𝜑· = (comp‘𝐶))
catidd.c (𝜑𝐶 ∈ Cat)
catidd.1 ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))
catidd.2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)
catidd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)
Assertion
Ref Expression
catidd (𝜑 → (Id‘𝐶) = (𝑥𝐵1 ))
Distinct variable groups:   𝑦,𝑓, 1   𝑥,𝐵   𝑥,𝑓,𝐶,𝑦   𝜑,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑦,𝑓)   · (𝑥,𝑦,𝑓)   1 (𝑥)   𝐻(𝑥,𝑦,𝑓)

Proof of Theorem catidd
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 catidd.2 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)
21ex 415 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥)) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
3 catidd.b . . . . . . . . . . . 12 (𝜑𝐵 = (Base‘𝐶))
43eleq2d 2898 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐶)))
53eleq2d 2898 . . . . . . . . . . 11 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐶)))
6 catidd.h . . . . . . . . . . . . 13 (𝜑𝐻 = (Hom ‘𝐶))
76oveqd 7167 . . . . . . . . . . . 12 (𝜑 → (𝑦𝐻𝑥) = (𝑦(Hom ‘𝐶)𝑥))
87eleq2d 2898 . . . . . . . . . . 11 (𝜑 → (𝑓 ∈ (𝑦𝐻𝑥) ↔ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)))
94, 5, 83anbi123d 1432 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))))
10 catidd.o . . . . . . . . . . . . 13 (𝜑· = (comp‘𝐶))
1110oveqd 7167 . . . . . . . . . . . 12 (𝜑 → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥))
1211oveqd 7167 . . . . . . . . . . 11 (𝜑 → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
1312eqeq1d 2823 . . . . . . . . . 10 (𝜑 → (( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
142, 9, 133imtr3d 295 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
15143expd 1349 . . . . . . . 8 (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))))
1615imp41 428 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
1716ralrimiva 3182 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
18 catidd.3 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)
1918ex 415 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
206oveqd 7167 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
2120eleq2d 2898 . . . . . . . . . . 11 (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
224, 5, 213anbi123d 1432 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))))
2310oveqd 7167 . . . . . . . . . . . 12 (𝜑 → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦))
2423oveqd 7167 . . . . . . . . . . 11 (𝜑 → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ))
2524eqeq1d 2823 . . . . . . . . . 10 (𝜑 → ((𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
2619, 22, 253imtr3d 295 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
27263expd 1349 . . . . . . . 8 (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))))
2827imp41 428 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)
2928ralrimiva 3182 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)
3017, 29jca 514 . . . . 5 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
3130ralrimiva 3182 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
32 catidd.1 . . . . . . . 8 ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))
3332ex 415 . . . . . . 7 (𝜑 → (𝑥𝐵1 ∈ (𝑥𝐻𝑥)))
346oveqd 7167 . . . . . . . 8 (𝜑 → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
3534eleq2d 2898 . . . . . . 7 (𝜑 → ( 1 ∈ (𝑥𝐻𝑥) ↔ 1 ∈ (𝑥(Hom ‘𝐶)𝑥)))
3633, 4, 353imtr3d 295 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝐶) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥)))
3736imp 409 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥))
38 eqid 2821 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
39 eqid 2821 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
40 eqid 2821 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
41 catidd.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
4241adantr 483 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
43 simpr 487 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4438, 39, 40, 42, 43catideu 16940 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ∃!𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
45 oveq1 7157 . . . . . . . . . 10 (𝑔 = 1 → (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
4645eqeq1d 2823 . . . . . . . . 9 (𝑔 = 1 → ((𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
4746ralbidv 3197 . . . . . . . 8 (𝑔 = 1 → (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
48 oveq2 7158 . . . . . . . . . 10 (𝑔 = 1 → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ))
4948eqeq1d 2823 . . . . . . . . 9 (𝑔 = 1 → ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
5049ralbidv 3197 . . . . . . . 8 (𝑔 = 1 → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
5147, 50anbi12d 632 . . . . . . 7 (𝑔 = 1 → ((∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)))
5251ralbidv 3197 . . . . . 6 (𝑔 = 1 → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)))
5352riota2 7133 . . . . 5 (( 1 ∈ (𝑥(Hom ‘𝐶)𝑥) ∧ ∃!𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓) ↔ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 ))
5437, 44, 53syl2anc 586 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓) ↔ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 ))
5531, 54mpbid 234 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 )
5655mpteq2dva 5153 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))) = (𝑥 ∈ (Base‘𝐶) ↦ 1 ))
57 eqid 2821 . . 3 (Id‘𝐶) = (Id‘𝐶)
5838, 39, 40, 41, 57cidfval 16941 . 2 (𝜑 → (Id‘𝐶) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))))
593mpteq1d 5147 . 2 (𝜑 → (𝑥𝐵1 ) = (𝑥 ∈ (Base‘𝐶) ↦ 1 ))
6056, 58, 593eqtr4d 2866 1 (𝜑 → (Id‘𝐶) = (𝑥𝐵1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  ∃!wreu 3140  cop 4566  cmpt 5138  cfv 6349  crio 7107  (class class class)co 7150  Basecbs 16477  Hom chom 16570  compcco 16571  Catccat 16929  Idccid 16930
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-cat 16933  df-cid 16934
This theorem is referenced by:  iscatd2  16946
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