Theorem fncnvima2 5679

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 5677	. . 3 |- ( F Fn A -> ( x e. ( `' F " B ) <-> ( x e. A /\ ( F ` x ) e. B ) ) )
2	1	abbi2dv 2312	. 2 |- ( F Fn A -> ( `' F " B ) = { x | ( x e. A /\ ( F ` x ) e. B ) } )
3		df-rab 2481	. 2 |- { x e. A | ( F ` x ) e. B } = { x | ( x e. A /\ ( F ` x ) e. B ) }
4	2, 3	eqtr4di 2244	1 |- ( F Fn A -> ( `' F " B ) = { x e. A | ( F ` x ) e. B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   `'ccnv 4658   "cima 4662   Fn wfn 5249   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  fniniseg2  5680  fnniniseg2  5681