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Theorem fssxp 6701
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:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))

Proof of Theorem fssxp
StepHypRef Expression
1 frel 6678 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 6225 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 6682 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 4005 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 17 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 6680 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 5653 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 585 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3959 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wss 3915   × cxp 5636  dom cdm 5638  ran crn 5639  Rel wrel 5643  wf 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-fun 6503  df-fn 6504  df-f 6505
This theorem is referenced by:  funssxp  6702  opelf  6708  dff2  7054  dff3  7055  fndifnfp  7127  fex2  7875  fabexg  7876  f2ndf  8057  f1o2ndf1  8059  mapex  8778  fsetsspwxp  8798  uniixp  8866  wdom2d  9523  rankfu  9820  dfac12lem2  10087  infmap2  10161  axdc3lem  10393  fnct  10480  tskcard  10724  ixxex  13282  imasvscafn  17426  imasvscaf  17428  fnmrc  17494  mrcfval  17495  isacs1i  17544  mreacs  17545  pjfval  21128  pjpm  21130  isngp2  23969  volf  24909  fgraphopab  41566  dfno2  41774  issmflem  45042
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