Theorem elimag 5080

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 4092	. . 3 |- ( y = A -> ( x B y <-> x B A ) )
2	1	rexbidv 2533	. 2 |- ( y = A -> ( E. x e. C x B y <-> E. x e. C x B A ) )
3		dfima2 5078	. 2 |- ( B " C ) = { y | E. x e. C x B y }
4	2, 3	elab2g 2953	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 2202   E.wrex 2511   class class class wbr 4088   "cima 4728

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  elima  5081  elrelimasn  5102  fvelima  5697  ecexr  6706