Theorem fssxp 5355

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 5342	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5124	. . 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 5343	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3196	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 14	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 5346	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4711	. . 3 |- ( ( dom F C_ A /\ ran F C_ B ) -> ( dom F X. ran F ) C_ ( A X. B ) )
9	6, 7, 8	syl2anc 409	. 2 |- ( F : A --> B -> ( dom F X. ran F ) C_ ( A X. B ) )
10	3, 9	sstrd 3152	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116   X. cxp 4602   dom cdm 4604   ran crn 4605   Rel wrel 4609   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  fex2  5356  funssxp  5357  opelf  5359  fabexg  5375  dff2  5629  dff3im  5630  f2ndf  6194  f1o2ndf1  6196  tfrlemibfn  6296  tfr1onlembfn  6312  tfrcllembfn  6325  mapex  6620  uniixp  6687  ixxex  9835  pw1nct  13883