Theorem fvimacnvi 5610

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 3724	. . 3 |- ( A e. ( `' F " B ) -> { A } C_ ( `' F " B ) )
2		funimass2 5276	. . 3 |- ( ( Fun F /\ { A } C_ ( `' F " B ) ) -> ( F " { A } ) C_ B )
3	1, 2	sylan2 284	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F " { A } ) C_ B )
4		cnvimass 4974	. . . . 5 |- ( `' F " B ) C_ dom F
5	4	sseli 3143	. . . 4 |- ( A e. ( `' F " B ) -> A e. dom F )
6		funfvex 5513	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
7		snssg 3716	. . . . 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 284	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B ) )
10		funfn 5228	. . . . . 6 |- ( Fun F <-> F Fn dom F )
11		fnsnfv 5555	. . . . . 6 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
12	10, 11	sylanb 282	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
13	5, 12	sylan2 284	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> { ( F ` A ) } = ( F " { A } ) )
14	13	sseq1d 3176	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( { ( F ` A ) } C_ B <-> ( F " { A } ) C_ B ) )
15	9, 14	bitrd 187	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> ( F " { A } ) C_ B ) )
16	3, 15	mpbird 166	1 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   {csn 3583   `'ccnv 4610   dom cdm 4611   "cima 4614   Fun wfun 5192   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  fvimacnv  5611  elpreima  5615