Theorem elimag 5079

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 4091	. . 3 |- ( y = A -> ( x B y <-> x B A ) )
2	1	rexbidv 2532	. 2 |- ( y = A -> ( E. x e. C x B y <-> E. x e. C x B A ) )
3		dfima2 5077	. 2 |- ( B " C ) = { y | E. x e. C x B y }
4	2, 3	elab2g 2952	1 |- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2201   E.wrex 2510   class class class wbr 4087   "cima 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-br 4088  df-opab 4150  df-xp 4730  df-cnv 4732  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737

This theorem is referenced by:  elima  5080  elrelimasn  5101  fvelima  5697  ecexr  6709