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Mirrors > Home > MPE Home > Th. List > fconst | Structured version Visualization version GIF version |
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 5302 | . . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fnmpti 6161 | . 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴 |
4 | rnxpss 5706 | . 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵} | |
5 | df-f 6034 | . 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵})) | |
6 | 3, 4, 5 | mpbir2an 690 | 1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 Vcvv 3351 ⊆ wss 3723 {csn 4317 × cxp 5248 ran crn 5251 Fn wfn 6025 ⟶wf 6026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-fun 6032 df-fn 6033 df-f 6034 |
This theorem is referenced by: fconstg 6233 fodomr 8271 ofsubeq0 11223 ser0f 13061 hashgval 13324 hashinf 13326 hashfxnn0 13328 hashfOLD 13330 prodf1f 14831 pwssplit1 19272 psrbag0 19709 xkofvcn 21708 ibl0 23773 dvcmul 23927 dvcmulf 23928 dvexp 23936 elqaalem3 24296 basellem7 25034 basellem9 25036 axlowdimlem8 26050 axlowdimlem9 26051 axlowdimlem10 26052 axlowdimlem11 26053 axlowdimlem12 26054 0oo 27984 occllem 28502 ho01i 29027 nlelchi 29260 hmopidmchi 29350 eulerpartlemt 30773 plymul02 30963 breprexpnat 31052 noetalem3 32202 fullfunfnv 32390 fullfunfv 32391 poimirlem16 33757 poimirlem19 33760 poimirlem23 33764 poimirlem24 33765 poimirlem25 33766 poimirlem28 33769 poimirlem29 33770 poimirlem30 33771 poimirlem31 33772 poimirlem32 33773 ftc1anclem5 33820 lfl0f 34876 diophrw 37846 pwssplit4 38183 ofsubid 39047 dvsconst 39053 dvsid 39054 binomcxplemnn0 39072 binomcxplemnotnn0 39079 aacllem 43073 |
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