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| Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) | 
| Ref | Expression | 
|---|---|
| fconst.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| fconst | ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fconst.1 | . . 3 ⊢ 𝐵 ∈ V | |
| 2 | fconstmpt 5746 | . . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 1, 2 | fnmpti 6710 | . 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴 | 
| 4 | rnxpss 6191 | . 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵} | |
| 5 | df-f 6564 | . 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵})) | |
| 6 | 3, 4, 5 | mpbir2an 711 | 1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 {csn 4625 × cxp 5682 ran crn 5685 Fn wfn 6555 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: fconstg 6794 fodomr 9169 fodomfir 9369 ofsubeq0 12264 ser0f 14097 hashgval 14373 hashinf 14375 hashfxnn0 14377 prodf1f 15929 pwssplit1 21059 psrbag0 22087 xkofvcn 23693 rrx0el 25433 ibl0 25823 dvcmul 25982 dvcmulf 25983 dvexp 25992 elqaalem3 26364 basellem7 27131 basellem9 27133 noetasuplem4 27782 axlowdimlem8 28965 axlowdimlem9 28966 axlowdimlem10 28967 axlowdimlem11 28968 axlowdimlem12 28969 0oo 30809 occllem 31323 ho01i 31848 nlelchi 32081 hmopidmchi 32171 elrgspnlem1 33247 eulerpartlemt 34374 plymul02 34562 breprexpnat 34650 fullfunfnv 35948 fullfunfv 35949 poimirlem16 37644 poimirlem19 37647 poimirlem23 37651 poimirlem24 37652 poimirlem25 37653 poimirlem28 37656 poimirlem29 37657 poimirlem30 37658 poimirlem31 37659 poimirlem32 37660 ftc1anclem5 37705 lfl0f 39071 diophrw 42775 pwssplit4 43106 ofsubid 44348 dvsconst 44354 dvsid 44355 binomcxplemnn0 44373 binomcxplemnotnn0 44380 functermc 49168 aacllem 49375 | 
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