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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 6132 | . 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 2030 {csn 4210 × cxp 5141 ⟶wf 5922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-fun 5928 df-fn 5929 df-f 5930 |
This theorem is referenced by: ramz 15776 psrlidm 19451 psrbag0 19542 00ply1bas 19658 ply1plusgfvi 19660 mbfpos 23463 i1f0 23499 axlowdimlem1 25867 axlowdimlem7 25873 axlowdim1 25884 hlim0 28220 0cnfn 28967 0lnfn 28972 circlemethnat 30847 circlevma 30848 noxp1o 31941 poimirlem29 33568 poimirlem30 33569 poimirlem31 33570 poimir 33572 broucube 33573 expgrowth 38851 |
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