Theorem fvimacnv 5749

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 5398 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 5746	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		funfvex 5643	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
3		opelcnvg 4901	. . . . . 6 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
4	2, 3	sylancom 420	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( <. ( F ` A ) , A >. e. `' F <-> <. A , ( F ` A ) >. e. F ) )
5	1, 4	mpbird 167	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> <. ( F ` A ) , A >. e. `' F )
6		elimasng 5095	. . . . 5 |- ( ( ( F ` A ) e. _V /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
7	2, 6	sylancom 420	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " { ( F ` A ) } ) <-> <. ( F ` A ) , A >. e. `' F ) )
8	5, 7	mpbird 167	. . 3 |- ( ( Fun F /\ A e. dom F ) -> A e. ( `' F " { ( F ` A ) } ) )
9		snssg 3801	. . . . . . . 8 |- ( ( F ` A ) e. _V -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
10	2, 9	syl 14	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
11		imass2 5103	. . . . . . 7 |- ( { ( F ` A ) } C_ B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
12	10, 11	biimtrdi 163	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) ) )
13	12	imp 124	. . . . 5 |- ( ( ( Fun F /\ A e. dom F ) /\ ( F ` A ) e. B ) -> ( `' F " { ( F ` A ) } ) C_ ( `' F " B ) )
14	13	sseld 3223	. . . 4 |- ( ( ( Fun F /\ A e. dom F ) /\ ( F ` A ) e. B ) -> ( A e. ( `' F " { ( F ` A ) } ) -> A e. ( `' F " B ) ) )
15	14	ex 115	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> ( A e. ( `' F " { ( F ` A ) } ) -> A e. ( `' F " B ) ) ) )
16	8, 15	mpid 42	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B -> A e. ( `' F " B ) ) )
17		fvimacnvi 5748	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )
18	17	ex 115	. . 3 |- ( Fun F -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
19	18	adantr 276	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( A e. ( `' F " B ) -> ( F ` A ) e. B ) )
20	16, 19	impbid 129	1 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {csn 3666   <.cop 3669   `'ccnv 4717   dom cdm 4718   "cima 4721   Fun wfun 5311   ` cfv 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  funimass3  5750  elpreima  5753  fisumss  11898  psrbaglesuppg  14630