Theorem fvimacnv 5994

Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5660 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
fvimacnv	|- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )

Proof of Theorem fvimacnv

Step	Hyp	Ref	Expression
1		funfvop 5991	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		fvex 5874	. . . . . . 7 |- ( F ` A ) e. _V
3		opelcnvg 5180	. . . . . . 7 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
4	2, 3	mpan 670	. . . . . 6 |- ( A e. dom F -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
5	4	adantl 466	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
6	1, 5	mpbird 232	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> <. ( F ` A ) , A >. e. `' F )
7		elimasng 5361	. . . . . 6 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
8	2, 7	mpan 670	. . . . 5 |- ( A e. dom F -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
9	8	adantl 466	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
10	6, 9	mpbird 232	. . 3 |- ( ( Fun F /\ A e. dom F ) -> A e. ( `' F " { ( F ` A ) } ) )
11	2	snss 4151	. . . . 5 |- ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B )
12		imass2 5370	. . . . 5 |- ( { ( F ` A ) } C_ B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
13	11, 12	sylbi 195	. . . 4 |- ( ( F ` A ) e. B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
14	13	sseld 3503	. . 3 |- ( ( F ` A ) e. B -> ( A e. ( `' F " { ( F ` A ) } ) -> A e. ( `' F " B ) ) )
15	10, 14	syl5com 30	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> A e. ( `' F " B ) ) )
16		fvimacnvi 5993	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )
17	16	ex 434	. . 3 |- ( Fun F -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
18	17	adantr 465	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
19	15, 18	impbid 191	1 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   _Vcvv 3113   C_ wss 3476   {csn 4027   <.cop 4033   `'ccnv 4998   dom cdm 4999   "cima 5002   Fun wfun 5580   ` cfv 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594

This theorem is referenced by:  funimass3  5995  elpreima  5999  iinpreima  6009  isr0  19970  rnelfmlem  20185  rnelfm  20186  fmfnfmlem2  20188  fmfnfmlem4  20190  fmfnfm  20191  metustidOLD  20794  metustid  20795  metustsymOLD  20796  metustsym  20797  metustexhalfOLD  20798  metustexhalf  20799  xppreima  27156  dstfrvel  28049  ballotlemrv  28095  grpokerinj  29948  arearect  30788  areaquad  30789  diaintclN  35855  dibintclN  35964  dihintcl  36141