Theorem fssxp 5530

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 5513	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5283	. . 3 |- ( Rel F -> F C_ ( dom F X. ran F ) )
3	1, 2	syl 14	. 2 |- ( F : A --> B -> F C_ ( dom F X. ran F ) )
4		fdm 5514	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3292	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 14	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 5517	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4857	. . 3 |- ( ( dom F C_ A /\ ran F C_ B ) -> ( dom F X. ran F ) C_ ( A X. B ) )
9	6, 7, 8	syl2anc 411	. 2 |- ( F : A --> B -> ( dom F X. ran F ) C_ ( A X. B ) )
10	3, 9	sstrd 3248	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3211   X. cxp 4747   dom cdm 4749   ran crn 4750   Rel wrel 4754   -->wf 5348

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  fex2  5531  funssxp  5532  opelf  5535  fabexg  5554  dff2  5821  dff3im  5822  f2ndf  6422  f1o2ndf1  6424  tfrlemibfn  6559  tfr1onlembfn  6575  tfrcllembfn  6588  mapex  6888  uniixp  6956  ixxex  10232  pw1nct  16777