Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5083 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
2 | ffn 5046 | . 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 1393 {csn 3375 × cxp 4343 Fn wfn 4897 ⟶wf 4898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: fconst2g 5376 |
Copyright terms: Public domain | W3C validator |