Theorem ab2rexex 7911

Description: Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7894. (Contributed by NM, 20-Sep-2011.)

Hypotheses

Ref	Expression
ab2rexex.1	⊢ 𝐴 ∈ V
ab2rexex.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
ab2rexex	⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V

Distinct variable groups:   𝑥,𝑧,𝐴   𝑦,𝑧,𝐵   𝑧,𝐶

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem ab2rexex

Step	Hyp	Ref	Expression
1		ab2rexex.1	. 2 ⊢ 𝐴 ∈ V
2		ab2rexex.2	. . 3 ⊢ 𝐵 ∈ V
3	2	abrexex 7894	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
4	1, 3	abrexex2 7901	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V

Syntax hints:   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3919  df-uni 4860  df-iun 4943

This theorem is referenced by:  plyval  26123  precsexlem4  28146  precsexlem5  28147  onmulscl  28209  zs12ex  28382  pstmfval  33904  pstmxmet  33905