Theorem opkelcnvkg 4250

Description: Kuratowski ordered pair membership in a Kuratowski converse. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
opkelcnvkg	|- ((A e. V /\ B e. W) -> (<<A, B>> e. `'kC <-> <<B, A>> e. C))

Proof of Theorem opkelcnvkg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnvk 4187	. 2 |- `'kC = {z | E.xE.y(z = <<x, y>> /\ <<y, x>> e. C)}
2		opkeq2 4061	. . 3 |- (x = A -> <<y, x>> = <<y, A>>)
3	2	eleq1d 2419	. 2 |- (x = A -> (<<y, x>> e. C <-> <<y, A>> e. C))
4		opkeq1 4060	. . 3 |- (y = B -> <<y, A>> = <<B, A>>)
5	4	eleq1d 2419	. 2 |- (y = B -> (<<y, A>> e. C <-> <<B, A>> e. C))
6	1, 3, 5	opkelopkabg 4246	1 |- ((A e. V /\ B e. W) -> (<<A, B>> e. `'kC <-> <<B, A>> e. C))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  <<copk 4058  `'_kccnvk 4176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-cnvk 4187

This theorem is referenced by:  opkelcnvk  4251  opkelcokg  4262