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Theorem catidd 17724
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 412 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥)) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
3 catidd.b . . . . . . . . . . . 12 (𝜑𝐵 = (Base‘𝐶))
43eleq2d 2826 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐶)))
53eleq2d 2826 . . . . . . . . . . 11 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐶)))
6 catidd.h . . . . . . . . . . . . 13 (𝜑𝐻 = (Hom ‘𝐶))
76oveqd 7449 . . . . . . . . . . . 12 (𝜑 → (𝑦𝐻𝑥) = (𝑦(Hom ‘𝐶)𝑥))
87eleq2d 2826 . . . . . . . . . . 11 (𝜑 → (𝑓 ∈ (𝑦𝐻𝑥) ↔ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)))
94, 5, 83anbi123d 1437 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))))
10 catidd.o . . . . . . . . . . . . 13 (𝜑· = (comp‘𝐶))
1110oveqd 7449 . . . . . . . . . . . 12 (𝜑 → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥))
1211oveqd 7449 . . . . . . . . . . 11 (𝜑 → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
1312eqeq1d 2738 . . . . . . . . . 10 (𝜑 → (( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
142, 9, 133imtr3d 293 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
15143expd 1353 . . . . . . . 8 (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))))
1615imp41 425 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
1716ralrimiva 3145 . . . . . 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 412 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
206oveqd 7449 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
2120eleq2d 2826 . . . . . . . . . . 11 (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
224, 5, 213anbi123d 1437 . . . . . . . . . 10 (𝜑 → ((𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))))
2310oveqd 7449 . . . . . . . . . . . 12 (𝜑 → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦))
2423oveqd 7449 . . . . . . . . . . 11 (𝜑 → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ))
2524eqeq1d 2738 . . . . . . . . . 10 (𝜑 → ((𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
2619, 22, 253imtr3d 293 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
27263expd 1353 . . . . . . . 8 (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))))
2827imp41 425 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)
2928ralrimiva 3145 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)
3017, 29jca 511 . . . . 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 3145 . . . 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 412 . . . . . . 7 (𝜑 → (𝑥𝐵1 ∈ (𝑥𝐻𝑥)))
346oveqd 7449 . . . . . . . 8 (𝜑 → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
3534eleq2d 2826 . . . . . . 7 (𝜑 → ( 1 ∈ (𝑥𝐻𝑥) ↔ 1 ∈ (𝑥(Hom ‘𝐶)𝑥)))
3633, 4, 353imtr3d 293 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝐶) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥)))
3736imp 406 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥))
38 eqid 2736 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
39 eqid 2736 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
40 eqid 2736 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
41 catidd.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
4241adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
43 simpr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
4438, 39, 40, 42, 43catideu 17719 . . . . 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 7439 . . . . . . . . . 10 (𝑔 = 1 → (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
4645eqeq1d 2738 . . . . . . . . 9 (𝑔 = 1 → ((𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
4746ralbidv 3177 . . . . . . . 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 7440 . . . . . . . . . 10 (𝑔 = 1 → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ))
4948eqeq1d 2738 . . . . . . . . 9 (𝑔 = 1 → ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
5049ralbidv 3177 . . . . . . . 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 3177 . . . . . 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 7414 . . . . 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 584 . . . 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 232 . . 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 5241 . 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 2736 . . 3 (Id‘𝐶) = (Id‘𝐶)
5838, 39, 40, 41, 57cidfval 17720 . 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 5236 . 2 (𝜑 → (𝑥𝐵1 ) = (𝑥 ∈ (Base‘𝐶) ↦ 1 ))
6056, 58, 593eqtr4d 2786 1 (𝜑 → (Id‘𝐶) = (𝑥𝐵1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  ∃!wreu 3377  cop 4631  cmpt 5224  cfv 6560  crio 7388  (class class class)co 7432  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-cat 17712  df-cid 17713
This theorem is referenced by:  iscatd2  17725
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