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| Mirrors > Home > MPE Home > Th. List > fconst6 | Structured version Visualization version GIF version | ||
| Description: A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.) |
| Ref | Expression |
|---|---|
| fconst6.1 | ⊢ 𝐵 ∈ 𝐶 |
| Ref | Expression |
|---|---|
| fconst6 | ⊢ (𝐴 × {𝐵}):𝐴⟶𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fconst6.1 | . 2 ⊢ 𝐵 ∈ 𝐶 | |
| 2 | fconst6g 6797 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 × {𝐵}):𝐴⟶𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 {csn 4626 × cxp 5683 ⟶wf 6557 |
| 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: ramz 17063 psrlidm 21982 psrbag0 22086 00ply1bas 22241 ply1plusgfvi 22243 mbfpos 25686 i1f0 25722 noxp1o 27708 axlowdimlem1 28957 axlowdimlem7 28963 axlowdim1 28974 hlim0 31254 0cnfn 31999 0lnfn 32004 elrgspnlem1 33246 circlemethnat 34656 circlevma 34657 poimirlem29 37656 poimirlem30 37657 poimirlem31 37658 poimir 37660 broucube 37661 expgrowth 44354 |
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