Theorem bj-projeq

Description: Substitution property for Proj . (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-projeq	⊢ ( 𝐴 = 𝐶 → ( 𝐵 = 𝐷 → ( 𝐴 Proj 𝐵 ) = ( 𝐶 Proj 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpr	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → 𝐵 = 𝐷 )
2		simpl	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → 𝐴 = 𝐶 )
3	2	sneqd	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → { 𝐴 } = { 𝐶 } )
4	1 3	imaeq12d	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐵 “ { 𝐴 } ) = ( 𝐷 “ { 𝐶 } ) )
5	4	eleq2d	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( { 𝑥 } ∈ ( 𝐵 “ { 𝐴 } ) ↔ { 𝑥 } ∈ ( 𝐷 “ { 𝐶 } ) ) )
6	5	abbidv	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → { 𝑥 ∣ { 𝑥 } ∈ ( 𝐵 “ { 𝐴 } ) } = { 𝑥 ∣ { 𝑥 } ∈ ( 𝐷 “ { 𝐶 } ) } )
7		df-bj-proj	⊢ ( 𝐴 Proj 𝐵 ) = { 𝑥 ∣ { 𝑥 } ∈ ( 𝐵 “ { 𝐴 } ) }
8		df-bj-proj	⊢ ( 𝐶 Proj 𝐷 ) = { 𝑥 ∣ { 𝑥 } ∈ ( 𝐷 “ { 𝐶 } ) }
9	6 7 8	3eqtr4g	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐴 Proj 𝐵 ) = ( 𝐶 Proj 𝐷 ) )
10	9	ex	⊢ ( 𝐴 = 𝐶 → ( 𝐵 = 𝐷 → ( 𝐴 Proj 𝐵 ) = ( 𝐶 Proj 𝐷 ) ) )