Theorem fssxp 6060

Description: A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fssxp	|- ( F : A --> B -> F C_ ( A X. B ) )

Proof of Theorem fssxp

Step	Hyp	Ref	Expression
1		frel 6050	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5656	. . 3 |- ( Rel F -> F C_ ( dom F X. ran F ) )
3	1, 2	syl 17	. 2 |- ( F : A --> B -> F C_ ( dom F X. ran F ) )
4		fdm 6051	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3657	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 17	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 6053	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 5225	. . 3 |- ( ( dom F C_ A /\ ran F C_ B ) -> ( dom F X. ran F ) C_ ( A X. B ) )
9	6, 7, 8	syl2anc 693	. 2 |- ( F : A --> B -> ( dom F X. ran F ) C_ ( A X. B ) )
10	3, 9	sstrd 3613	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   -->wf 5884

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  funssxp  6061  opelf  6065  dff2  6371  dff3  6372  fndifnfp  6442  fex2  7121  fabexg  7122  f2ndf  7283  f1o2ndf1  7285  mapex  7863  uniixp  7931  hartogslem1  8447  wdom2d  8485  rankfu  8740  dfac12lem2  8966  infmap2  9040  axdc3lem  9272  fnct  9359  tskcard  9603  dfle2  11980  ixxex  12186  imasvscafn  16197  imasvscaf  16199  fnmrc  16267  mrcfval  16268  isacs1i  16318  mreacs  16319  pjfval  20050  pjpm  20052  hausdiag  21448  isngp2  22401  volf  23297  fgraphopab  37788  issmflem  40936