MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fssxp Structured version   Visualization version   GIF version

Theorem fssxp 6762
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 6740 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 6287 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 6744 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 4041 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 17 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 6742 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 5699 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 584 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3993 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wss 3950   × cxp 5682  dom cdm 5684  ran crn 5685  Rel wrel 5689  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by:  funssxp  6763  opelf  6768  dff2  7118  dff3  7119  fndifnfp  7197  fex2  7959  fabexd  7960  fabexgOLD  7962  f2ndf  8146  f1o2ndf1  8148  mapexOLD  8873  fsetsspwxp  8894  uniixp  8962  wdom2d  9621  rankfu  9918  dfac12lem2  10186  infmap2  10258  axdc3lem  10491  fnct  10578  tskcard  10822  ixxex  13399  imasvscafn  17583  imasvscaf  17585  fnmrc  17651  mrcfval  17652  isacs1i  17701  mreacs  17702  pjfval  21727  pjpm  21729  isngp2  24611  volf  25565  fgraphopab  43220  dfno2  43446  issmflem  46747
  Copyright terms: Public domain W3C validator