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Theorem fssxp 5089
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 5080 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
2 relssdmrn 4871 . . 3 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 14 . 2 (𝐹:𝐴𝐵𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4 fdm 5081 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5 eqimss 3052 . . . 4 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
64, 5syl 14 . . 3 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
7 frn 5083 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
8 xpss12 4473 . . 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 3010 1 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wss 2974   × cxp 4369  dom cdm 4371  ran crn 4372  Rel wrel 4376  wf 4928
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936
This theorem is referenced by:  fex2  5090  funssxp  5091  opelf  5093  fabexg  5108  dff2  5343  dff3im  5344  f2ndf  5878  f1o2ndf1  5880  tfrlemibfn  5977  tfr1onlembfn  5993  tfrcllembfn  6006  ixxex  8998
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