Theorem vxp 38998

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 38997	. . 3 ⊢ (𝐴 × V) = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴}
2	1	cnveqi 5858	. 2 ⊢ ◡(𝐴 × V) = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴}
3		cnvxp 6152	. 2 ⊢ ◡(𝐴 × V) = (V × 𝐴)
4		cnvopab 6135	. 2 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}
5	2, 3, 4	3eqtr3i 2793	1 ⊢ (V × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝐴}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3453  {copab 5171   × cxp 5657  ^◡ccnv 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667

This theorem is used by: (None)