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Metamath Proof Explorer

Theorem catcid

Description: The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses catccatid.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
catccatid.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
catcid.o ⊢ 1 = ( Id ‘ 𝐶 )
catcid.i ⊢ 𝐼 = ( idfunc ‘ 𝑋 )
catcid.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
catcid.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion catcid ( 𝜑 → ( 1 ‘ 𝑋 ) = 𝐼 )

Proof

Step Hyp Ref Expression
1 catccatid.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2 catccatid.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 catcid.o ⊢ 1 = ( Id ‘ 𝐶 )
4 catcid.i ⊢ 𝐼 = ( idfunc ‘ 𝑋 )
5 catcid.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
6 catcid.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
7 1 2 catccatid ⊢ ( 𝑈 ∈ 𝑉 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ ( idfunc ‘ 𝑥 ) ) ) )
8 5 7 syl ⊢ ( 𝜑 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ ( idfunc ‘ 𝑥 ) ) ) )
9 8 simprd ⊢ ( 𝜑 → ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ ( idfunc ‘ 𝑥 ) ) )
10 3 9 eqtrid ⊢ ( 𝜑 → 1 = ( 𝑥 ∈ 𝐵 ↦ ( idfunc ‘ 𝑥 ) ) )
11 simpr ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑥 = 𝑋 )
12 11 fveq2d ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( idfunc ‘ 𝑥 ) = ( idfunc ‘ 𝑋 ) )
13 fvexd ⊢ ( 𝜑 → ( idfunc ‘ 𝑋 ) ∈ V )
14 10 12 6 13 fvmptd ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( idfunc ‘ 𝑋 ) )
15 14 4 eqtr4di ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = 𝐼 )