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Theorem fssxp 6310
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 6296 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 5910 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 6299 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 3875 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 17 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 6297 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 5370 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 579 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3830 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wss 3791   × cxp 5353  dom cdm 5355  ran crn 5356  Rel wrel 5360  wf 6131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-fun 6137  df-fn 6138  df-f 6139
This theorem is referenced by:  funssxp  6311  opelf  6315  dff2  6635  dff3  6636  fndifnfp  6709  fex2  7400  fabexg  7401  f2ndf  7564  f1o2ndf1  7566  mapex  8146  uniixp  8217  wdom2d  8774  rankfu  9037  dfac12lem2  9301  infmap2  9375  axdc3lem  9607  fnct  9694  tskcard  9938  ixxex  12498  imasvscafn  16583  imasvscaf  16585  fnmrc  16653  mrcfval  16654  isacs1i  16703  mreacs  16704  pjfval  20449  pjpm  20451  isngp2  22809  volf  23733  fgraphopab  38729  issmflem  41845
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