Theorem sikeq 4241

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

Assertion

Ref	Expression
sikeq	|- (A = B -> SIk A = SIk B)

Proof of Theorem sikeq

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

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . . . 7 |- (A = B -> (<<w, t>> e. A <-> <<w, t>> e. B))
2	1	3anbi3d 1258	. . . . . 6 |- (A = B -> ((y = {w} /\ z = {t} /\ <<w, t>> e. A) <-> (y = {w} /\ z = {t} /\ <<w, t>> e. B)))
3	2	2exbidv 1628	. . . . 5 |- (A = B -> (E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. A) <-> E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. B)))
4	3	anbi2d 684	. . . 4 |- (A = B -> ((x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. A)) <-> (x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. B))))
5	4	2exbidv 1628	. . 3 |- (A = B -> (E.yE.z(x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. A)) <-> E.yE.z(x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. B))))
6	5	abbidv 2467	. 2 |- (A = B -> {x | E.yE.z(x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. A))} = {x | E.yE.z(x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. B))})
7		df-sik 4192	. 2 |- SIk A = {x | E.yE.z(x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. A))}
8		df-sik 4192	. 2 |- SIk B = {x | E.yE.z(x = <<y, z>> /\ E.wE.t(y = {w} /\ z = {t} /\ <<w, t>> e. B))}
9	6, 7, 8	3eqtr4g 2410	1 |- (A = B -> SIk A = SIk B)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  {csn 3737  <<copk 4057   SI_k csik 4181

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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sik 4192

This theorem is referenced by:  sikeqi  4242  sikeqd  4243  imagekeq  4244  sikexg  4296