Theorem fconstg 6715

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 4589	. . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵})
2	1	xpeq2d 5653	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵}))
3		feq1 6634	. . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥}))
4		feq3 6636	. . . 4 ⊢ ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
5	3, 4	sylan9bb 509	. . 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 3442	. . 3 ⊢ 𝑥 ∈ V
8	7	fconst 6714	. 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥}
9	6, 8	vtoclg 3511	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {csn 4579   × cxp 5621  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by:  fnconstg  6716  fconst6g  6717  xpsng  7077  fvconst2g  7142  fconst2g  7143  symgpssefmnd  19293  xkoptsub  23557  mbfconstlem  25544  i1fmulclem  25619  i1fmulc  25620  itg2mulclem  25663  dvcmulf  25864  dvef  25900  coemulc  26176  resf1o  32686  locfinref  33807  ccatmulgnn0dir  34509  frlmvscadiccat  42479