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Theorem fssxp 5178
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 5165 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 4951 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 14 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 5166 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 3078 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 14 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 5169 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 4545 . . 3 ((dom 𝐹𝐴 ∧ ran 𝐹𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
96, 7, 8syl2anc 403 . 2 (𝐹:𝐴𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵))
103, 9sstrd 3035 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wss 2999   × cxp 4436  dom cdm 4438  ran crn 4439  Rel wrel 4443  wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019
This theorem is referenced by:  fex2  5179  funssxp  5180  opelf  5182  fabexg  5198  dff2  5443  dff3im  5444  f2ndf  5991  f1o2ndf1  5993  tfrlemibfn  6093  tfr1onlembfn  6109  tfrcllembfn  6122  mapex  6409  ixxex  9315
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