Theorem fvimacnv 6332

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 5972 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 6329	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		fvex 6201	. . . . . . 7 |- ( F ` A ) e. _V
3		opelcnvg 5302	. . . . . . 7 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
4	2, 3	mpan 706	. . . . . 6 |- ( A e. dom F -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
5	4	adantl 482	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
6	1, 5	mpbird 247	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> <. ( F ` A ) , A >. e. `' F )
7		elimasng 5491	. . . . . 6 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
8	2, 7	mpan 706	. . . . 5 |- ( A e. dom F -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
9	8	adantl 482	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
10	6, 9	mpbird 247	. . 3 |- ( ( Fun F /\ A e. dom F ) -> A e. ( `' F " { ( F ` A ) } ) )
11	2	snss 4316	. . . . 5 |- ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B )
12		imass2 5501	. . . . 5 |- ( { ( F ` A ) } C_ B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
13	11, 12	sylbi 207	. . . 4 |- ( ( F ` A ) e. B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
14	13	sseld 3602	. . 3 |- ( ( F ` A ) e. B -> ( A e. ( `' F " { ( F ` A ) } ) -> A e. ( `' F " B ) ) )
15	10, 14	syl5com 31	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> A e. ( `' F " B ) ) )
16		fvimacnvi 6331	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )
17	16	ex 450	. . 3 |- ( Fun F -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
18	17	adantr 481	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
19	15, 18	impbid 202	1 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   `'ccnv 5113   dom cdm 5114   "cima 5117   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  funimass3  6333  elpreima  6337  iinpreima  6345  isr0  21540  rnelfmlem  21756  rnelfm  21757  fmfnfmlem2  21759  fmfnfmlem4  21761  fmfnfm  21762  metustid  22359  metustsym  22360  metustexhalf  22361  xppreima  29449  dstfrvel  30535  ballotlemrv  30581  grpokerinj  33692  diaintclN  36347  dibintclN  36456  dihintcl  36633  arearect  37801  areaquad  37802