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Theorem fconstg 5412
Description: A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
Assertion
Ref Expression
fconstg (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})

Proof of Theorem fconstg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 3603 . . . 4 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21xpeq2d 4650 . . 3 (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵}))
3 feq1 5348 . . . 4 ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥}))
4 feq3 5350 . . . 4 ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
53, 4sylan9bb 462 . . 3 (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
62, 1, 5syl2anc 411 . 2 (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
7 vex 2740 . . 3 𝑥 ∈ V
87fconst 5411 . 2 (𝐴 × {𝑥}):𝐴⟶{𝑥}
96, 8vtoclg 2797 1 (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {csn 3592   × cxp 4624  wf 5212
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220
This theorem is referenced by:  fnconstg  5413  fconst6g  5414  xpsng  5691  fvconst2g  5730  fconst2g  5731  dvef  14081
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