Theorem elimag 4692

Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Assertion

Ref	Expression
elimag	|- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) )

Distinct variable groups:   x, A   x, B   x, C

Allowed substitution hint:   V( x)

Proof of Theorem elimag

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3789	. . 3 |- ( y = A -> ( x B y <-> x B A ) )
2	1	rexbidv 2369	. 2 |- ( y = A -> ( E. x e. C x B y <-> E. x e. C x B A ) )
3		dfima2 4690	. 2 |- ( B " C ) = { y | E. x e. C x B y }
4	2, 3	elab2g 2740	1 |- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  elima  4693  fvelima  5246  ecexr  6134