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Theorem fssxp 5405
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 5392 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 5170 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 14 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 5393 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 3224 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 14 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 5396 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 4754 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 411 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3180 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wss 3144   × cxp 4645  dom cdm 4647  ran crn 4648  Rel wrel 4652  wf 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-dm 4657  df-rn 4658  df-fun 5240  df-fn 5241  df-f 5242
This theorem is referenced by:  fex2  5406  funssxp  5407  opelf  5409  fabexg  5425  dff2  5684  dff3im  5685  f2ndf  6255  f1o2ndf1  6257  tfrlemibfn  6357  tfr1onlembfn  6373  tfrcllembfn  6386  mapex  6684  uniixp  6751  ixxex  9935  pw1nct  15239
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