Theorem elpmg

Description: The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013)

	Ref	Expression
Assertion	elpmg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( Fun 𝐶 ∧ 𝐶 ⊆ ( 𝐵 × 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pmvalg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ↑pm 𝐵 ) = { 𝑔 ∈ 𝒫 ( 𝐵 × 𝐴 ) ∣ Fun 𝑔 } )
2	1	eleq2d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ 𝐶 ∈ { 𝑔 ∈ 𝒫 ( 𝐵 × 𝐴 ) ∣ Fun 𝑔 } ) )
3		funeq	⊢ ( 𝑔 = 𝐶 → ( Fun 𝑔 ↔ Fun 𝐶 ) )
4	3	elrab	⊢ ( 𝐶 ∈ { 𝑔 ∈ 𝒫 ( 𝐵 × 𝐴 ) ∣ Fun 𝑔 } ↔ ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ∧ Fun 𝐶 ) )
5	2 4	syl6bb	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ∧ Fun 𝐶 ) ) )
6	5	biancomd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( Fun 𝐶 ∧ 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ) ) )
7		elex	⊢ ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) → 𝐶 ∈ V )
8	7	a1i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) → 𝐶 ∈ V ) )
9		xpexg	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐵 × 𝐴 ) ∈ V )
10	9	ancoms	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 × 𝐴 ) ∈ V )
11		ssexg	⊢ ( ( 𝐶 ⊆ ( 𝐵 × 𝐴 ) ∧ ( 𝐵 × 𝐴 ) ∈ V ) → 𝐶 ∈ V )
12	11	expcom	⊢ ( ( 𝐵 × 𝐴 ) ∈ V → ( 𝐶 ⊆ ( 𝐵 × 𝐴 ) → 𝐶 ∈ V ) )
13	10 12	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ⊆ ( 𝐵 × 𝐴 ) → 𝐶 ∈ V ) )
14		elpwg	⊢ ( 𝐶 ∈ V → ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ↔ 𝐶 ⊆ ( 𝐵 × 𝐴 ) ) )
15	14	a1i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ V → ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ↔ 𝐶 ⊆ ( 𝐵 × 𝐴 ) ) ) )
16	8 13 15	pm5.21ndd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ↔ 𝐶 ⊆ ( 𝐵 × 𝐴 ) ) )
17	16	anbi2d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( Fun 𝐶 ∧ 𝐶 ∈ 𝒫 ( 𝐵 × 𝐴 ) ) ↔ ( Fun 𝐶 ∧ 𝐶 ⊆ ( 𝐵 × 𝐴 ) ) ) )
18	6 17	bitrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐶 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( Fun 𝐶 ∧ 𝐶 ⊆ ( 𝐵 × 𝐴 ) ) ) )