Theorem fssxp 4575

Description: A mapping is a class of ordered pairs. (The proof was 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		fdm 4567	. . . 4 |- (F:A-->B -> dom F = A)
2		eqimss 2665	. . . 4 |- (dom F = A -> dom F C_ A)
3	1, 2	syl 12	. . 3 |- (F:A-->B -> dom F C_ A)
4		frn 4569	. . 3 |- (F:A-->B -> ran F C_ B)
5		xpss12 4089	. . 3 |- ((dom F C_ A /\ ran F C_ B) -> (dom F X. ran F) C_ (A X. B))
6	3, 4, 5	syl11anc 524	. 2 |- (F:A-->B -> (dom F X. ran F) C_ (A X. B))
7		frel 4566	. . 3 |- (F:A-->B -> Rel F)
8		relssdmrn 4416	. . 3 |- (Rel F -> F C_ (dom F X. ran F))
9		sstr2 2623	. . 3 |- (F C_ (dom F X. ran F) -> ((dom F X. ran F) C_ (A X. B) -> F C_ (A X. B)))
10	7, 8, 9	3syl 24	. 2 |- (F:A-->B -> ((dom F X. ran F) C_ (A X. B) -> F C_ (A X. B)))
11	6, 10	mpd 29	1 |- (F:A-->B -> F C_ (A X. B))

Syntax hints:   -> wi 3   = wceq 1298   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991  -->wf 3994

This theorem is referenced by:  funssxp 4577  opelf 4579  fabexg 4596  dff2 4789  dff3 4790  fopabssxp 4797  mapex 5387  mapval2 5394  mapsspw 5400  uniixp 5416  infmap2 8850  lmbrf 9208  iscauf 9217  iscau5 9219  iscaunns 9222  lmclimnn 9242  h2hcau 10481  h2hlm 10482  scprefat 14380  1alg 15069  fnctartar 15284  fnctartar2 15285  tlmval 15903  heiborlem33 15987

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

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