Theorem fconst 5497

Description: A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Hypothesis

Ref	Expression
fconst.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
fconst	⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Proof of Theorem fconst

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconst.1	. . 3 ⊢ 𝐵 ∈ V
2		fconstmpt 4743	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 5428	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 5136	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 5298	. 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
6	3, 4, 5	mpbir2an 947	1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Syntax hints:   ∈ wcel 2180  Vcvv 2779   ⊆ wss 3177  {csn 3646   × cxp 4694  ran crn 4697   Fn wfn 5289  ⟶wf 5290

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-fun 5296  df-fn 5297  df-f 5298

This theorem is referenced by:  fconstg  5498  exmidfodomrlemim  7347  ser0f  10723  prodf1f  12020  dvexp  15350