Metamath Proof Explorer

Theorem cidfval

Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses cidfval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
cidfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
cidfval.o ⊢ · = ( comp ‘ 𝐶 )
cidfval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
cidfval.i ⊢ 1 = ( Id ‘ 𝐶 )
Assertion cidfval ( 𝜑 → 1 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )

Proof

Step Hyp Ref Expression
1 cidfval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
2 cidfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
3 cidfval.o ⊢ · = ( comp ‘ 𝐶 )
4 cidfval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
5 cidfval.i ⊢ 1 = ( Id ‘ 𝐶 )
6 fvexd ⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) ∈ V )
7 fveq2 ⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = ( Base ‘ 𝐶 ) )
8 7 1 eqtr4di ⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = 𝐵 )
9 fvexd ⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑐 ) ∈ V )
10 simpl ⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → 𝑐 = 𝐶 )
11 10 fveq2d ⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑐 ) = ( Hom ‘ 𝐶 ) )
12 11 2 eqtr4di ⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑐 ) = 𝐻 )
13 fvexd ⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( comp ‘ 𝑐 ) ∈ V )
14 simpll ⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → 𝑐 = 𝐶 )
15 14 fveq2d ⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( comp ‘ 𝑐 ) = ( comp ‘ 𝐶 ) )
16 15 3 eqtr4di ⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( comp ‘ 𝑐 ) = · )
17 simpllr ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → 𝑏 = 𝐵 )
18 simplr ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ℎ = 𝐻 )
19 18 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( 𝑥 ℎ 𝑥 ) = ( 𝑥 𝐻 𝑥 ) )
20 18 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( 𝑦 ℎ 𝑥 ) = ( 𝑦 𝐻 𝑥 ) )
21 simpr ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → 𝑜 = · )
22 21 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) = ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) )
23 22 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) )
24 23 eqeq1d ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ↔ ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ) )
25 20 24 raleqbidv ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ) )
26 18 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( 𝑥 ℎ 𝑦 ) = ( 𝑥 𝐻 𝑦 ) )
27 21 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) = ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) )
28 27 oveqd ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) )
29 28 eqeq1d ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ↔ ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) )
30 26 29 raleqbidv ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) )
31 25 30 anbi12d ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) )
32 17 31 raleqbidv ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) )
33 19 32 riotaeqbidv ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( ℩ 𝑔 ∈ ( 𝑥 ℎ 𝑥 ) ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ) = ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) )
34 17 33 mpteq12dv ⊢ ( ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) ∧ 𝑜 = · ) → ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑔 ∈ ( 𝑥 ℎ 𝑥 ) ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ) ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
35 13 16 34 csbied2 ⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ⦋ ( comp ‘ 𝑐 ) / 𝑜 ⦌ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑔 ∈ ( 𝑥 ℎ 𝑥 ) ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ) ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
36 9 12 35 csbied2 ⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ⦋ ( Hom ‘ 𝑐 ) / ℎ ⦌ ⦋ ( comp ‘ 𝑐 ) / 𝑜 ⦌ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑔 ∈ ( 𝑥 ℎ 𝑥 ) ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ) ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
37 6 8 36 csbied2 ⊢ ( 𝑐 = 𝐶 → ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ⦋ ( Hom ‘ 𝑐 ) / ℎ ⦌ ⦋ ( comp ‘ 𝑐 ) / 𝑜 ⦌ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑔 ∈ ( 𝑥 ℎ 𝑥 ) ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ) ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
38 df-cid ⊢ Id = ( 𝑐 ∈ Cat ↦ ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ⦋ ( Hom ‘ 𝑐 ) / ℎ ⦌ ⦋ ( comp ‘ 𝑐 ) / 𝑜 ⦌ ( 𝑥 ∈ 𝑏 ↦ ( ℩ 𝑔 ∈ ( 𝑥 ℎ 𝑥 ) ∀ 𝑦 ∈ 𝑏 ( ∀ 𝑓 ∈ ( 𝑦 ℎ 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ 𝑜 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ 𝑜 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
39 37 38 1 mptfvmpt ⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
40 4 39 syl ⊢ ( 𝜑 → ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )
41 5 40 eqtrid ⊢ ( 𝜑 → 1 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑔 ∈ ( 𝑥 𝐻 𝑥 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ · 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) )