Theorem fvimacnvi 6331

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 4339	. . 3 |- ( A e. ( `' F " B ) -> { A } C_ ( `' F " B ) )
2		funimass2 5972	. . 3 |- ( ( Fun F /\ { A } C_ ( `' F " B ) ) -> ( F " { A } ) C_ B )
3	1, 2	sylan2 491	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F " { A } ) C_ B )
4		fvex 6201	. . . 4 |- ( F ` A ) e. _V
5	4	snss 4316	. . 3 |- ( ( F ` A ) e. B <-> { ( F ` A ) } C_ B )
6		cnvimass 5485	. . . . . 6 |- ( `' F " B ) C_ dom F
7	6	sseli 3599	. . . . 5 |- ( A e. ( `' F " B ) -> A e. dom F )
8		funfn 5918	. . . . . 6 |- ( Fun F <-> F Fn dom F )
9		fnsnfv 6258	. . . . . 6 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
10	8, 9	sylanb 489	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
11	7, 10	sylan2 491	. . . 4 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> { ( F ` A ) } = ( F " { A } ) )
12	11	sseq1d 3632	. . 3 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( { ( F ` A ) } C_ B <-> ( F " { A } ) C_ B ) )
13	5, 12	syl5bb 272	. 2 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( ( F ` A ) e. B <-> ( F " { A } ) C_ B ) )
14	3, 13	mpbird 247	1 |- ( ( Fun F /\ A e. ( `' F " B ) ) -> ( F ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   `'ccnv 5113   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` 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:  fvimacnv  6332  elpreima  6337  iinpreima  6345  lmhmpreima  19048  mpfind  19536  ofco2  20257  carsggect  30380