Theorem elima3 5128

Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Hypothesis

Ref	Expression
elima.1	|- A e. _V

Assertion

Ref	Expression
elima3	|- ( A e. ( B " C ) <-> E. x ( x e. C /\ <. x , A >. e. B ) )

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

Proof of Theorem elima3

Step	Hyp	Ref	Expression
1		elima.1	. . 3 |- A e. _V
2	1	elima2 5127	. 2 |- ( A e. ( B " C ) <-> E. x ( x e. C /\ x B A ) )
3		df-br 4126	. . . 4 |- ( x B A <-> <. x , A >. e. B )
4	3	anbi2i 461	. . 3 |- ( ( x e. C /\ x B A ) <-> ( x e. C /\ <. x , A >. e. B ) )
5	4	exbii 1658	. 2 |- ( E. x ( x e. C /\ x B A ) <-> E. x ( x e. C /\ <. x , A >. e. B ) )
6	2, 5	bitri 184	1 |- ( A e. ( B " C ) <-> E. x ( x e. C /\ <. x , A >. e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   class class class wbr 4125   "cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  cnvresima  5272  imaiun  5956