Theorem fssxp 5558

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 5541	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5311	. . 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 5542	. . . 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 5545	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4884	. . 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 4774   dom cdm 4776   ran crn 4777   Rel wrel 4781   -->wf 5376

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 4251  ax-pow 4313  ax-pr 4348

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 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-dm 4786  df-rn 4787  df-fun 5382  df-fn 5383  df-f 5384

This theorem is used by:  fex2  5559  funssxp  5560  opelf  5563  fabexg  5582  dff2  5855  dff3im  5856  f2ndf  6466  f1o2ndf1  6468  tfrlemibfn  6603  tfr1onlembfn  6619  tfrcllembfn  6632  mapex  6932  fsetsspwxp  6952  uniixp  7007  ixxex  10318  pw1nct  17238