Theorem elpreima 5583

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 4946	. . . . 5 |- ( `' F " C ) C_ dom F
2	1	sseli 3124	. . . 4 |- ( B e. ( `' F " C ) -> B e. dom F )
3		fndm 5266	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	eleq2d 2227	. . . 4 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
5	2, 4	syl5ib 153	. . 3 |- ( F Fn A -> ( B e. ( `' F " C ) -> B e. A ) )
6		fnfun 5264	. . . . 5 |- ( F Fn A -> Fun F )
7		fvimacnvi 5578	. . . . 5 |- ( ( Fun F /\ B e. ( `' F " C ) ) -> ( F ` B ) e. C )
8	6, 7	sylan 281	. . . 4 |- ( ( F Fn A /\ B e. ( `' F " C ) ) -> ( F ` B ) e. C )
9	8	ex 114	. . 3 |- ( F Fn A -> ( B e. ( `' F " C ) -> ( F ` B ) e. C ) )
10	5, 9	jcad 305	. 2 |- ( F Fn A -> ( B e. ( `' F " C ) -> ( B e. A /\ ( F ` B ) e. C ) ) )
11		fvimacnv 5579	. . . . 5 |- ( ( Fun F /\ B e. dom F ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
12	11	funfni 5267	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
13	12	biimpd 143	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) e. C -> B e. ( `' F " C ) ) )
14	13	expimpd 361	. 2 |- ( F Fn A -> ( ( B e. A /\ ( F ` B ) e. C ) -> B e. ( `' F " C ) ) )
15	10, 14	impbid 128	1 |- ( F Fn A -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2128   `'ccnv 4582   dom cdm 4583   "cima 4586   Fun wfun 5161   Fn wfn 5162   ` cfv 5167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-fv 5175

This theorem is referenced by:  fniniseg  5584  fncnvima2  5585  rexsupp  5588  unpreima  5589  respreima  5592  fisumss  11271  fprodssdc  11469  tanvalap  11587  cncnpi  12588  cncnp  12590  cnpdis  12602  tx1cn  12629  tx2cn  12630  txcnmpt  12633  txdis1cn  12638  xmeterval  12795  cnbl0  12894  cnblcld  12895