Theorem fssxp 5555

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 5538	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5308	. . 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 5539	. . . 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 5542	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4882	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220   X. cxp 4772   dom cdm 4774   ran crn 4775   Rel wrel 4779   -->wf 5373

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381

This theorem is used by:  fex2  5556  funssxp  5557  opelf  5560  fabexg  5579  dff2  5852  dff3im  5853  f2ndf  6462  f1o2ndf1  6464  tfrlemibfn  6599  tfr1onlembfn  6615  tfrcllembfn  6628  mapex  6928  fsetsspwxp  6948  uniixp  7003  ixxex  10301  pw1nct  17033