Theorem elpreima 5418

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 4795	. . . . 5 |- ( `' F " C ) C_ dom F
2	1	sseli 3021	. . . 4 |- ( B e. ( `' F " C ) -> B e. dom F )
3		fndm 5113	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	eleq2d 2157	. . . 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 5111	. . . . 5 |- ( F Fn A -> Fun F )
7		fvimacnvi 5413	. . . . 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 5414	. . . . 5 |- ( ( Fun F /\ B e. dom F ) -> ( ( F ` B ) e. C <-> B e. ( `' F " C ) ) )
12	11	funfni 5114	. . . 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 1438   `'ccnv 4437   dom cdm 4438   "cima 4441   Fun wfun 5009   Fn wfn 5010   ` cfv 5015

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023

This theorem is referenced by:  fniniseg  5419  fncnvima2  5420  rexsupp  5423  unpreima  5424  respreima  5427  fisumss  10784  tanvalap  10999