Theorem cnvkeq 4216

Description: Equality theorem for Kuratowski converse. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
cnvkeq	|- (A = B -> `'kA = `'kB)

Proof of Theorem cnvkeq

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

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . 5 |- (A = B -> (<<z, y>> e. A <-> <<z, y>> e. B))
2	1	anbi2d 684	. . . 4 |- (A = B -> ((x = <<y, z>> /\ <<z, y>> e. A) <-> (x = <<y, z>> /\ <<z, y>> e. B)))
3	2	2exbidv 1628	. . 3 |- (A = B -> (E.yE.z(x = <<y, z>> /\ <<z, y>> e. A) <-> E.yE.z(x = <<y, z>> /\ <<z, y>> e. B)))
4	3	abbidv 2468	. 2 |- (A = B -> {x | E.yE.z(x = <<y, z>> /\ <<z, y>> e. A)} = {x | E.yE.z(x = <<y, z>> /\ <<z, y>> e. B)})
5		df-cnvk 4187	. 2 |- `'kA = {x | E.yE.z(x = <<y, z>> /\ <<z, y>> e. A)}
6		df-cnvk 4187	. 2 |- `'kB = {x | E.yE.z(x = <<y, z>> /\ <<z, y>> e. B)}
7	4, 5, 6	3eqtr4g 2410	1 |- (A = B -> `'kA = `'kB)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  <<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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-cnvk 4187

This theorem is referenced by:  cnvkeqi  4217  cnvkeqd  4218  cokeq2  4232  cnvkexg  4287