Theorem fssxp 5395

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 5382	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5161	. . 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 5383	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3221	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 14	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 5386	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4745	. . 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 3177	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1363   C_ wss 3141   X. cxp 4636   dom cdm 4638   ran crn 4639   Rel wrel 4643   -->wf 5224

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232

This theorem is referenced by:  fex2  5396  funssxp  5397  opelf  5399  fabexg  5415  dff2  5673  dff3im  5674  f2ndf  6241  f1o2ndf1  6243  tfrlemibfn  6343  tfr1onlembfn  6359  tfrcllembfn  6372  mapex  6668  uniixp  6735  ixxex  9913  pw1nct  15049