Theorem elpreima 5762

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 5097	. . . . 5 |- ( `' F " C ) C_ dom F
2	1	sseli 3221	. . . 4 |- ( B e. ( `' F " C ) -> B e. dom F )
3		fndm 5426	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	eleq2d 2299	. . . 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 5424	. . . . 5 |- ( F Fn A -> Fun F )
7		fvimacnvi 5757	. . . . 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 5758	. . . . 5 |- ( ( Fun F /\ B e. dom F ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
12	11	funfni 5429	. . . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   `'ccnv 4722   dom cdm 4723   "cima 4726   Fun wfun 5318   Fn wfn 5319   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  fniniseg  5763  fncnvima2  5764  rexsupp  5767  unpreima  5768  respreima  5771  fisumss  11943  fprodssdc  12141  tanvalap  12259  1arith  12930  ghmpreima  13843  ghmnsgpreima  13846  kerf1ghm  13851  psrbaglesuppg  14676  psrbagfi  14677  cncnpi  14942  cncnp  14944  cnpdis  14956  tx1cn  14983  tx2cn  14984  txcnmpt  14987  txdis1cn  14992  xmeterval  15149  cnbl0  15248  cnblcld  15249