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Theorem fconst 5190
Description: A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Hypothesis
Ref Expression
fconst.1 𝐵 ∈ V
Assertion
Ref Expression
fconst (𝐴 × {𝐵}):𝐴⟶{𝐵}

Proof of Theorem fconst
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fconst.1 . . 3 𝐵 ∈ V
2 fconstmpt 4473 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2fnmpti 5128 . 2 (𝐴 × {𝐵}) Fn 𝐴
4 rnxpss 4849 . 2 ran (𝐴 × {𝐵}) ⊆ {𝐵}
5 df-f 5006 . 2 ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
63, 4, 5mpbir2an 888 1 (𝐴 × {𝐵}):𝐴⟶{𝐵}
Colors of variables: wff set class
Syntax hints:  wcel 1438  Vcvv 2619  wss 2997  {csn 3441   × cxp 4426  ran crn 4429   Fn wfn 4997  wf 4998
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 3949  ax-pow 4001  ax-pr 4027
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 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-f 5006
This theorem is referenced by:  fconstg  5191  exmidfodomrlemim  6806  iser0f  9913  ser0f  9915
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