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Mirrors > Home > ILE Home > Th. List > fnconstg | GIF version |
Description: A cross 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 5242 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
2 | ffn 5195 | . 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 {csn 3466 × cxp 4465 Fn wfn 5044 ⟶wf 5045 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-fun 5051 df-fn 5052 df-f 5053 |
This theorem is referenced by: fconst2g 5551 nninfsellemeqinf 12617 |
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