Theorem fssxp 5453

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 5440	. . 3 |- ( F : A --> B -> Rel F )
2		relssdmrn 5212	. . 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 5441	. . . 4 |- ( F : A --> B -> dom F = A )
5		eqimss 3251	. . . 4 |- ( dom F = A -> dom F C_ A )
6	4, 5	syl 14	. . 3 |- ( F : A --> B -> dom F C_ A )
7		frn 5444	. . 3 |- ( F : A --> B -> ran F C_ B )
8		xpss12 4790	. . 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 3207	1 |- ( F : A --> B -> F C_ ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3170   X. cxp 4681   dom cdm 4683   ran crn 4684   Rel wrel 4688   -->wf 5276

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-dm 4693  df-rn 4694  df-fun 5282  df-fn 5283  df-f 5284

This theorem is referenced by:  fex2  5454  funssxp  5455  opelf  5458  fabexg  5475  dff2  5737  dff3im  5738  f2ndf  6325  f1o2ndf1  6327  tfrlemibfn  6427  tfr1onlembfn  6443  tfrcllembfn  6456  mapex  6754  uniixp  6821  ixxex  10041  pw1nct  16081