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Theorem fconstg 6730
Description: A Cartesian 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 4597 . . . 4 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21xpeq2d 5664 . . 3 (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵}))
3 feq1 6650 . . . 4 ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥}))
4 feq3 6652 . . . 4 ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
53, 4sylan9bb 511 . . 3 (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
62, 1, 5syl2anc 585 . 2 (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
7 vex 3450 . . 3 𝑥 ∈ V
87fconst 6729 . 2 (𝐴 × {𝑥}):𝐴⟶{𝑥}
96, 8vtoclg 3526 1 (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {csn 4587   × cxp 5632  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  fnconstg  6731  fconst6g  6732  xpsng  7086  fvconst2g  7152  fconst2g  7153  symgpssefmnd  19178  xkoptsub  23008  mbfconstlem  24994  i1fmulclem  25070  i1fmulc  25071  itg2mulclem  25114  dvcmulf  25312  dvef  25347  coemulc  25619  resf1o  31650  locfinref  32425  ccatmulgnn0dir  33157  frlmvscadiccat  40684
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