Theorem fvopab3ig

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999)

	Ref	Expression
Hypotheses	fvopab3ig.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	fvopab3ig.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	fvopab3ig.3	⊢ ( 𝑥 ∈ 𝐶 → ∃* 𝑦 𝜑 )
	fvopab3ig.4	⊢ 𝐹 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) }
Assertion	fvopab3ig	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝜒 → ( 𝐹 ‘ 𝐴 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fvopab3ig.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		fvopab3ig.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3		fvopab3ig.3	⊢ ( 𝑥 ∈ 𝐶 → ∃* 𝑦 𝜑 )
4		fvopab3ig.4	⊢ 𝐹 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) }
5		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
6	5 1	anbi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) ↔ ( 𝐴 ∈ 𝐶 ∧ 𝜓 ) ) )
7	2	anbi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝐶 ∧ 𝜓 ) ↔ ( 𝐴 ∈ 𝐶 ∧ 𝜒 ) ) )
8	6 7	opelopabg	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ↔ ( 𝐴 ∈ 𝐶 ∧ 𝜒 ) ) )
9	8	biimpar	⊢ ( ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ ( 𝐴 ∈ 𝐶 ∧ 𝜒 ) ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } )
10	9	exp43	⊢ ( 𝐴 ∈ 𝐶 → ( 𝐵 ∈ 𝐷 → ( 𝐴 ∈ 𝐶 → ( 𝜒 → ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ) ) ) )
11	10	pm2.43a	⊢ ( 𝐴 ∈ 𝐶 → ( 𝐵 ∈ 𝐷 → ( 𝜒 → ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ) ) )
12	11	imp	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝜒 → ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ) )
13	4	fveq1i	⊢ ( 𝐹 ‘ 𝐴 ) = ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ‘ 𝐴 )
14		funopab	⊢ ( Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ↔ ∀ 𝑥 ∃* 𝑦 ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) )
15		moanimv	⊢ ( ∃* 𝑦 ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐶 → ∃* 𝑦 𝜑 ) )
16	3 15	mpbir	⊢ ∃* 𝑦 ( 𝑥 ∈ 𝐶 ∧ 𝜑 )
17	14 16	mpgbir	⊢ Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) }
18		funopfv	⊢ ( Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } → ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ‘ 𝐴 ) = 𝐵 ) )
19	17 18	ax-mp	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } → ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } ‘ 𝐴 ) = 𝐵 )
20	13 19	syl5eq	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝜑 ) } → ( 𝐹 ‘ 𝐴 ) = 𝐵 )
21	12 20	syl6	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝜒 → ( 𝐹 ‘ 𝐴 ) = 𝐵 ) )