Theorem elimag 5086

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 4097	. . 3 |- ( y = A -> ( x B y <-> x B A ) )
2	1	rexbidv 2534	. 2 |- ( y = A -> ( E. x e. C x B y <-> E. x e. C x B A ) )
3		dfima2 5084	. 2 |- ( B " C ) = { y | E. x e. C x B y }
4	2, 3	elab2g 2954	1 |- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093   "cima 4734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  elima  5087  elrelimasn  5109  fvelima  5706  ecexr  6750