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Theorem fconstg 5287
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 3506 . . . 4 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21xpeq2d 4531 . . 3 (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵}))
3 feq1 5223 . . . 4 ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥}))
4 feq3 5225 . . . 4 ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
53, 4sylan9bb 455 . . 3 (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
62, 1, 5syl2anc 406 . 2 (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
7 vex 2661 . . 3 𝑥 ∈ V
87fconst 5286 . 2 (𝐴 × {𝑥}):𝐴⟶{𝑥}
96, 8vtoclg 2718 1 (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wcel 1463  {csn 3495   × cxp 4505  wf 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095
This theorem is referenced by:  fnconstg  5288  fconst6g  5289  xpsng  5561  fvconst2g  5600  fconst2g  5601  dvef  12762
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