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Theorem fssxp 6027
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 6017 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 5625 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 6018 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 3642 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 17 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 6020 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 5196 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 692 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3598 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wss 3560   × cxp 5082  dom cdm 5084  ran crn 5085  Rel wrel 5089  wf 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861
This theorem is referenced by:  funssxp  6028  opelf  6032  dff2  6337  dff3  6338  fndifnfp  6407  fex2  7083  fabexg  7084  f2ndf  7243  f1o2ndf1  7245  mapex  7823  uniixp  7891  hartogslem1  8407  wdom2d  8445  rankfu  8700  dfac12lem2  8926  infmap2  9000  axdc3lem  9232  fnct  9319  tskcard  9563  dfle2  11940  ixxex  12144  imasvscafn  16137  imasvscaf  16139  fnmrc  16207  mrcfval  16208  isacs1i  16258  mreacs  16259  pjfval  19990  pjpm  19992  hausdiag  21388  isngp2  22341  volf  23237  fgraphopab  37308  issmflem  40273
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