Theorem fvimacnv 5633

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 5296 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 5630	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		funfvex 5534	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
3		opelcnvg 4809	. . . . . 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 4998	. . . . 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 3728	. . . . . . . 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 5006	. . . . . . 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 3156	. . . 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 5632	. . . 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 2148   _Vcvv 2739   C_ wss 3131   {csn 3594   <.cop 3597   `'ccnv 4627   dom cdm 4628   "cima 4631   Fun wfun 5212   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  funimass3  5634  elpreima  5637  fisumss  11402