Theorem fssxp 5550

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 5533	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5303	. . 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 5534	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3302	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 14	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 5537	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4877	. . 3 |- ( ( dom F C_ A /\ ran F C_ B ) -> ( dom F X. ran F ) C_ ( A X. B ) )
9	6, 7, 8	syl2anc 415	. 2 |- ( F : A --> B -> ( dom F X. ran F ) C_ ( A X. B ) )
10	3, 9	sstrd 3258	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220   X. cxp 4767   dom cdm 4769   ran crn 4770   Rel wrel 4774   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  fex2  5551  funssxp  5552  opelf  5555  fabexg  5574  dff2  5843  dff3im  5844  f2ndf  6452  f1o2ndf1  6454  tfrlemibfn  6589  tfr1onlembfn  6605  tfrcllembfn  6618  mapex  6918  fsetsspwxp  6938  uniixp  6993  ixxex  10280  pw1nct  16947