Metamath Proof Explorer


Theorem fabexg

Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008)

Ref Expression
Hypothesis fabexg.1 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴𝐵𝜑 ) }
Assertion fabexg ( ( 𝐴𝐶𝐵𝐷 ) → 𝐹 ∈ V )

Proof

Step Hyp Ref Expression
1 fabexg.1 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴𝐵𝜑 ) }
2 xpexg ( ( 𝐴𝐶𝐵𝐷 ) → ( 𝐴 × 𝐵 ) ∈ V )
3 pwexg ( ( 𝐴 × 𝐵 ) ∈ V → 𝒫 ( 𝐴 × 𝐵 ) ∈ V )
4 fssxp ( 𝑥 : 𝐴𝐵𝑥 ⊆ ( 𝐴 × 𝐵 ) )
5 velpw ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 × 𝐵 ) )
6 4 5 sylibr ( 𝑥 : 𝐴𝐵𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) )
7 6 anim1i ( ( 𝑥 : 𝐴𝐵𝜑 ) → ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) )
8 7 ss2abi { 𝑥 ∣ ( 𝑥 : 𝐴𝐵𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) }
9 1 8 eqsstri 𝐹 ⊆ { 𝑥 ∣ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) }
10 ssab2 { 𝑥 ∣ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } ⊆ 𝒫 ( 𝐴 × 𝐵 )
11 9 10 sstri 𝐹 ⊆ 𝒫 ( 𝐴 × 𝐵 )
12 ssexg ( ( 𝐹 ⊆ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝒫 ( 𝐴 × 𝐵 ) ∈ V ) → 𝐹 ∈ V )
13 11 12 mpan ( 𝒫 ( 𝐴 × 𝐵 ) ∈ V → 𝐹 ∈ V )
14 2 3 13 3syl ( ( 𝐴𝐶𝐵𝐷 ) → 𝐹 ∈ V )