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

Theorem fssxp 6098
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 6088 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 5694 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 6089 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 3690 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 17 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 6091 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 5158 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 694 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3646 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wss 3607   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  funssxp  6099  opelf  6103  dff2  6411  dff3  6412  fndifnfp  6483  fex2  7163  fabexg  7164  f2ndf  7328  f1o2ndf1  7330  mapex  7905  uniixp  7973  hartogslem1  8488  wdom2d  8526  rankfu  8778  dfac12lem2  9004  infmap2  9078  axdc3lem  9310  fnct  9397  tskcard  9641  dfle2  12018  ixxex  12224  imasvscafn  16244  imasvscaf  16246  fnmrc  16314  mrcfval  16315  isacs1i  16365  mreacs  16366  pjfval  20098  pjpm  20100  hausdiag  21496  isngp2  22448  volf  23343  fgraphopab  38105  issmflem  41257
  Copyright terms: Public domain W3C validator