Theorem fconstg 6661

Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Assertion

Ref	Expression
fconstg	⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})

Proof of Theorem fconstg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4571	. . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	xpeq2d 5619	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵}))
3		feq1 6581	. . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥}))
4		feq3 6583	. . . 4 ⊢ ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
5	3, 4	sylan9bb 510	. . 3 ⊢ (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
6	2, 1, 5	syl2anc 584	. 2 ⊢ (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
7		vex 3436	. . 3 ⊢ 𝑥 ∈ V
8	7	fconst 6660	. 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥}
9	6, 8	vtoclg 3505	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {csn 4561   × cxp 5587  ⟶wf 6429

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437

This theorem is referenced by:  fnconstg  6662  fconst6g  6663  xpsng  7011  fvconst2g  7077  fconst2g  7078  symgpssefmnd  19003  xkoptsub  22805  mbfconstlem  24791  i1fmulclem  24867  i1fmulc  24868  itg2mulclem  24911  dvcmulf  25109  dvef  25144  coemulc  25416  resf1o  31065  locfinref  31791  ccatmulgnn0dir  32521  frlmvscadiccat  40237