Theorem vxp 38645

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 38644	. . 3 ⊢ (𝐴 × V) = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴}
2	1	cnveqi 5819	. 2 ⊢ ◡(𝐴 × V) = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴}
3		cnvxp 6112	. 2 ⊢ ◡(𝐴 × V) = (V × 𝐴)
4		cnvopab 6094	. 2 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}
5	2, 3, 4	3eqtr3i 2772	1 ⊢ (V × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}

Syntax hints:   = wceq 1548   ∈ wcel 2121  Vcvv 3433  {copab 5137   × cxp 5619  ^◡ccnv 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629

This theorem is referenced by: (None)