Theorem imakeq1 4225

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

Assertion

Ref	Expression
imakeq1	|- (A = B -> (A"kC) = (B"kC))

Proof of Theorem imakeq1

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

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 |- (A = B -> (<<y, x>> e. A <-> <<y, x>> e. B))
2	1	rexbidv 2636	. . 3 |- (A = B -> (E.y e. C <<y, x>> e. A <-> E.y e. C <<y, x>> e. B))
3	2	abbidv 2468	. 2 |- (A = B -> {x | E.y e. C <<y, x>> e. A} = {x | E.y e. C <<y, x>> e. B})
4		df-imak 4190	. 2 |- (A"kC) = {x | E.y e. C <<y, x>> e. A}
5		df-imak 4190	. 2 |- (B"kC) = {x | E.y e. C <<y, x>> e. B}
6	3, 4, 5	3eqtr4g 2410	1 |- (A = B -> (A"kC) = (B"kC))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  <<copk 4058  "_kcimak 4180

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-rex 2621  df-imak 4190

This theorem is referenced by:  imakeq1i  4227  imakeq1d  4229