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Theorem fconst2 5734
Description: A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
Hypothesis
Ref Expression
fvconst2.1 𝐵 ∈ V
Assertion
Ref Expression
fconst2 (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))

Proof of Theorem fconst2
StepHypRef Expression
1 fvconst2.1 . 2 𝐵 ∈ V
2 fconst2g 5732 . 2 (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
31, 2ax-mp 5 1 (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  Vcvv 2738  {csn 3593   × cxp 4625  wf 5213
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225
This theorem is referenced by:  map1  6812
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