Theorem fncnvima2 5309

Description: Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fncnvima2	|- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } )

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

Proof of Theorem fncnvima2

Step	Hyp	Ref	Expression
1		elpreima 5307	. . 3 |- ( F Fn A -> ( x e. ( `' F " B ) <-> ( x e. A /\ ( F ` x ) e. B ) ) )
2	1	abbi2dv 2197	. 2 |- ( F Fn A -> ( `' F " B ) = { x | ( x e. A /\ ( F ` x ) e. B ) } )
3		df-rab 2357	. 2 |- { x e. A | ( F ` x ) e. B } = { x | ( x e. A /\ ( F ` x ) e. B ) }
4	2, 3	syl6eqr 2131	1 |- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352   `'ccnv 4362   "cima 4366   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  fniniseg2  5310  fnniniseg2  5311