Theorem elimag 5071

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 4086	. . 3 |- ( y = A -> ( x B y <-> x B A ) )
2	1	rexbidv 2531	. 2 |- ( y = A -> ( E. x e. C x B y <-> E. x e. C x B A ) )
3		dfima2 5069	. 2 |- ( B " C ) = { y | E. x e. C x B y }
4	2, 3	elab2g 2950	1 |- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082   "cima 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731

This theorem is referenced by:  elima  5072  elrelimasn  5093  fvelima  5684  ecexr  6683