Theorem imakeq2 4225

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

Assertion

Ref	Expression
imakeq2	|- (A = B -> (C"kA) = (C"kB))

Proof of Theorem imakeq2

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

Step	Hyp	Ref	Expression
1		rexeq 2808	. . 3 |- (A = B -> (E.y e. A <<y, x>> e. C <-> E.y e. B <<y, x>> e. C))
2	1	abbidv 2467	. 2 |- (A = B -> {x | E.y e. A <<y, x>> e. C} = {x | E.y e. B <<y, x>> e. C})
3		df-imak 4189	. 2 |- (C"kA) = {x | E.y e. A <<y, x>> e. C}
4		df-imak 4189	. 2 |- (C"kB) = {x | E.y e. B <<y, x>> e. C}
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> (C"kA) = (C"kB))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  <<copk 4057  "_kcimak 4179

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-or 359  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-nfc 2478  df-rex 2620  df-imak 4189

This theorem is referenced by:  imakeq2i  4227  imakeq2d  4229  addceq1  4383  phieq  4570  opeq1  4578  opeq2  4579  proj1eq  4589  proj2eq  4590