Theorem elpreima 5307

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 4708	. . . . 5 |- ( `' F " C ) C_ dom F
2	1	sseli 2995	. . . 4 |- ( B e. ( `' F " C ) -> B e. dom F )
3		fndm 5018	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	eleq2d 2148	. . . 4 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
5	2, 4	syl5ib 152	. . 3 |- ( F Fn A -> ( B e. ( `' F " C ) -> B e. A ) )
6		fnfun 5016	. . . . 5 |- ( F Fn A -> Fun F )
7		fvimacnvi 5302	. . . . 5 |- ( ( Fun F /\ B e. ( `' F " C ) ) -> ( F ` B ) e. C )
8	6, 7	sylan 277	. . . 4 |- ( ( F Fn A /\ B e. ( `' F " C ) ) -> ( F ` B ) e. C )
9	8	ex 113	. . 3 |- ( F Fn A -> ( B e. ( `' F " C ) -> ( F ` B ) e. C ) )
10	5, 9	jcad 301	. 2 |- ( F Fn A -> ( B e. ( `' F " C ) -> ( B e. A /\ ( F ` B ) e. C ) ) )
11		fvimacnv 5303	. . . . 5 |- ( ( Fun F /\ B e. dom F ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
12	11	funfni 5019	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
13	12	biimpd 142	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) e. C -> B e. ( `' F " C ) ) )
14	13	expimpd 355	. 2 |- ( F Fn A -> ( ( B e. A /\ ( F ` B ) e. C ) -> B e. ( `' F " C ) ) )
15	10, 14	impbid 127	1 |- ( F Fn A -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   `'ccnv 4362   dom cdm 4363   "cima 4366   Fun wfun 4916   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-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:  fniniseg  5308  fncnvima2  5309  rexsupp  5312  unpreima  5313  respreima  5316