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| Mirrors > Home > MPE Home > Th. List > fnconstg | Structured version Visualization version GIF version | ||
| Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.) |
| Ref | Expression |
|---|---|
| fnconstg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fconstg 6795 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
| 2 | 1 | ffnd 6737 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 {csn 4626 × cxp 5683 Fn wfn 6556 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: fconst2g 7223 ofc1 7725 ofc2 7726 caofid0l 7730 caofid0r 7731 caofid1 7732 caofid2 7733 fnsuppres 8216 fczsupp0 8218 fczfsuppd 9426 brwdom2 9613 cantnf0 9715 ofnegsub 12264 ofsubge0 12265 pwsplusgval 17535 pwsmulrval 17536 pwsvscafval 17539 pwsco1mhm 18845 dprdsubg 20044 pwsmgp 20324 pwssplit1 21058 frlmpwsfi 21772 frlmbas 21775 frlmvscaval 21788 islindf4 21858 psrascl 21999 tmdgsum2 24104 0plef 25707 0pledm 25708 itg1ge0 25721 mbfi1fseqlem5 25754 xrge0f 25766 itg2ge0 25770 itg2addlem 25793 bddibl 25875 dvidlem 25950 rolle 26028 dveq0 26039 dv11cn 26040 tdeglem4 26099 mdeg0 26109 fta1blem 26210 qaa 26365 basellem9 27132 noextendseq 27712 noetainflem4 27785 fdifsuppconst 32698 elrspunidl 33456 ofcc 34107 ofcof 34108 eulerpartlemt 34373 matunitlindflem1 37623 matunitlindflem2 37624 ptrecube 37627 poimirlem1 37628 poimirlem2 37629 poimirlem3 37630 poimirlem4 37631 poimirlem5 37632 poimirlem6 37633 poimirlem7 37634 poimirlem10 37637 poimirlem11 37638 poimirlem12 37639 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 poimirlem22 37649 poimirlem23 37650 poimirlem28 37655 poimirlem29 37656 poimirlem31 37658 poimirlem32 37659 broucube 37661 cnpwstotbnd 37804 eqlkr2 39101 fsuppssind 42603 pwssplit4 43101 mpaaeu 43162 rngunsnply 43181 ofoaid1 43371 ofoaid2 43372 naddcnffo 43377 ofdivrec 44345 dvconstbi 44353 zlmodzxzscm 48273 aacllem 49320 |
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