Theorem elpreima 5831

Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
elpreima	|- ( F Fn A -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )

Proof of Theorem elpreima

Step	Hyp	Ref	Expression
1		cnvimass 5153	. . . . 5 |- ( `' F " C ) C_ dom F
2	1	sseli 3244	. . . 4 |- ( B e. ( `' F " C ) -> B e. dom F )
3		fndm 5483	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	eleq2d 2308	. . . 4 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
5	2, 4	imbitrid 154	. . 3 |- ( F Fn A -> ( B e. ( `' F " C ) -> B e. A ) )
6		fnfun 5481	. . . . 5 |- ( F Fn A -> Fun F )
7		fvimacnvi 5826	. . . . 5 |- ( ( Fun F /\ B e. ( `' F " C ) ) -> ( F ` B ) e. C )
8	6, 7	sylan 283	. . . 4 |- ( ( F Fn A /\ B e. ( `' F " C ) ) -> ( F ` B ) e. C )
9	8	ex 115	. . 3 |- ( F Fn A -> ( B e. ( `' F " C ) -> ( F ` B ) e. C ) )
10	5, 9	jcad 307	. 2 |- ( F Fn A -> ( B e. ( `' F " C ) -> ( B e. A /\ ( F ` B ) e. C ) ) )
11		fvimacnv 5827	. . . . 5 |- ( ( Fun F /\ B e. dom F ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
12	11	funfni 5486	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
13	12	biimpd 144	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) e. C -> B e. ( `' F " C ) ) )
14	13	expimpd 363	. 2 |- ( F Fn A -> ( ( B e. A /\ ( F ` B ) e. C ) -> B e. ( `' F " C ) ) )
15	10, 14	impbid 129	1 |- ( F Fn A -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   `'ccnv 4775   dom cdm 4776   "cima 4779   Fun wfun 5374   Fn wfn 5375   ` cfv 5380

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348

This proof 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-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388

This theorem is used by:  fniniseg  5832  fncnvima2  5833  unpreima  5836  respreima  5839  suppimacnvfn  6490  suppcofn  6510  fisumss  12184  fprodssdc  12382  tanvalap  12500  1arith  13175  ghmpreima  14128  ghmnsgpreima  14131  kerf1ghm  14136  psrbaglesuppg  15166  psrbagfi  15168  psrbaglecl  15169  psrbagcon  15171  cncnpi  15454  cncnp  15456  cnpdis  15468  tx1cn  15495  tx2cn  15496  txcnmpt  15499  txdis1cn  15504  xmeterval  15661  cnbl0  15760  cnblcld  15761