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| Mirrors > Home > MPE Home > Th. List > fnmptd | Structured version Visualization version GIF version | ||
| Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| fnmptd.1 | ⊢ Ⅎ𝑥𝜑 | 
| fnmptd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | 
| fnmptd.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | 
| Ref | Expression | 
|---|---|
| fnmptd | ⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnmptd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | fnmptd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
| 3 | 2 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)) | 
| 4 | 1, 3 | ralrimi 3256 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) | 
| 5 | fnmptd.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 6 | 5 | fnmpt 6707 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) | 
| 7 | 4, 6 | syl 17 | 1 ⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 ∀wral 3060 ↦ cmpt 5224 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-fun 6562 df-fn 6563 | 
| This theorem is referenced by: rngqiprngimf1 21311 elrgspnsubrunlem2 33253 nsgmgc 33441 nsgqusf1o 33445 elrspunsn 33458 ply1gsumz 33620 ply1degltdimlem 33674 evls1fldgencl 33721 ply1annidllem 33745 algextdeglem6 33764 metakunt33 42239 limsupequzmptlem 45748 liminfval2 45788 smflimmpt 46830 smflimsuplem7 46846 cfsetsnfsetfo 47077 | 
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