Theorem fssxp 5749

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 5740	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5534	. . 3 |- ( Rel F -> F C_ ( dom F X. ran F ) )
3	1, 2	syl 16	. 2 |- ( F : A --> B -> F C_ ( dom F X. ran F ) )
4		fdm 5741	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3551	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 16	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 5743	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 5117	. . 3 |- ( ( dom F C_ A /\ ran F C_ B ) -> ( dom F X. ran F ) C_ ( A X. B ) )
9	6, 7, 8	syl2anc 661	. 2 |- ( F : A --> B -> ( dom F X. ran F ) C_ ( A X. B ) )
10	3, 9	sstrd 3509	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3471   X. cxp 5006   dom cdm 5008   ran crn 5009   Rel wrel 5013   -->wf 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598

This theorem is referenced by:  funssxp  5750  opelf  5753  dff2  6044  dff3  6045  fndifnfp  6101  fex2  6754  fabexg  6755  f2ndf  6905  f1o2ndf1  6907  mapex  7444  uniixp  7511  hartogslem1  7985  wdom2d  8024  rankfu  8312  dfac12lem2  8541  infmap2  8615  axdc3lem  8847  tskcard  9176  dfle2  11378  ixxex  11565  imasvscafn  14954  imasvscaf  14956  fnmrc  15024  mrcfval  15025  isacs1i  15074  mreacs  15075  pjfval  18864  pjpm  18866  hausdiag  20272  isngp2  21243  volf  22066  fnct  27693  fgraphopab  31374