Theorem fvimacnvi 5717

Description: A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Assertion

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

Proof of Theorem fvimacnvi

Step	Hyp	Ref	Expression
1		snssi 3788	. . 3 |- ( A e. ( `' F " B ) -> { A } C_ ( `' F " B ) )
2		funimass2 5371	. . 3 |- ( ( Fun F /\ { A } C_ ( `' F " B ) ) -> ( F " { A } ) C_ B )
3	1, 2	sylan2 286	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F " { A } ) C_ B )
4		cnvimass 5064	. . . . 5 |- ( `' F " B ) C_ dom F
5	4	sseli 3197	. . . 4 |- ( A e. ( `' F " B ) -> A e. dom F )
6		funfvex 5616	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
7		snssg 3778	. . . . 5 |- ( ( F ` A ) e. _V -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
8	6, 7	syl 14	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
9	5, 8	sylan2 286	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
10		funfn 5320	. . . . . 6 |- ( Fun F <-> F Fn dom F )
11		fnsnfv 5661	. . . . . 6 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
12	10, 11	sylanb 284	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
13	5, 12	sylan2 286	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> { ( F ` A ) } = ( F " { A } ) )
14	13	sseq1d 3230	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( { ( F ` A ) } C_ B <-> ( F " { A } ) C_ B ) )
15	9, 14	bitrd 188	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> ( F " { A } ) C_ B ) )
16	3, 15	mpbird 167	1 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   {csn 3643   `'ccnv 4692   dom cdm 4693   "cima 4696   Fun wfun 5284   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  fvimacnv  5718  elpreima  5722  psrbaglesuppg  14549