Theorem vxp 38584

Description: Cartesian product of the universe and a class. (Contributed by Peter Mazsa, 3-Dec-2020.)

Assertion

Ref	Expression
vxp	⊢ (V × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem vxp

Step	Hyp	Ref	Expression
1		xpv 38583	. . 3 ⊢ (𝐴 × V) = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴}
2	1	cnveqi 5829	. 2 ⊢ ◡(𝐴 × V) = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴}
3		cnvxp 6121	. 2 ⊢ ◡(𝐴 × V) = (V × 𝐴)
4		cnvopab 6100	. 2 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}
5	2, 3, 4	3eqtr3i 2767	1 ⊢ (V × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {copab 5147   × cxp 5629  ^◡ccnv 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639

This theorem is referenced by: (None)