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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 6570 | . 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 2114 {csn 4569 × cxp 5555 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: ramz 16363 psrlidm 20185 psrbag0 20276 00ply1bas 20410 ply1plusgfvi 20412 mbfpos 24254 i1f0 24290 axlowdimlem1 26730 axlowdimlem7 26736 axlowdim1 26747 hlim0 29014 0cnfn 29759 0lnfn 29764 circlemethnat 31914 circlevma 31915 noxp1o 33172 poimirlem29 34923 poimirlem30 34924 poimirlem31 34925 poimir 34927 broucube 34928 expgrowth 40674 |
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