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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 5751 | . . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fnmpti 6712 | . 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴 |
4 | rnxpss 6194 | . 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵} | |
5 | df-f 6567 | . 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 2106 Vcvv 3478 ⊆ wss 3963 {csn 4631 × cxp 5687 ran crn 5690 Fn wfn 6558 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: fconstg 6796 fodomr 9167 fodomfir 9366 ofsubeq0 12261 ser0f 14093 hashgval 14369 hashinf 14371 hashfxnn0 14373 prodf1f 15925 pwssplit1 21076 psrbag0 22104 xkofvcn 23708 rrx0el 25446 ibl0 25837 dvcmul 25996 dvcmulf 25997 dvexp 26006 elqaalem3 26378 basellem7 27145 basellem9 27147 noetasuplem4 27796 axlowdimlem8 28979 axlowdimlem9 28980 axlowdimlem10 28981 axlowdimlem11 28982 axlowdimlem12 28983 0oo 30818 occllem 31332 ho01i 31857 nlelchi 32090 hmopidmchi 32180 elrgspnlem1 33232 eulerpartlemt 34353 plymul02 34540 breprexpnat 34628 fullfunfnv 35928 fullfunfv 35929 poimirlem16 37623 poimirlem19 37626 poimirlem23 37630 poimirlem24 37631 poimirlem25 37632 poimirlem28 37635 poimirlem29 37636 poimirlem30 37637 poimirlem31 37638 poimirlem32 37639 ftc1anclem5 37684 lfl0f 39051 diophrw 42747 pwssplit4 43078 ofsubid 44320 dvsconst 44326 dvsid 44327 binomcxplemnn0 44345 binomcxplemnotnn0 44352 aacllem 49032 |
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