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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmptd2f | Structured version Visualization version GIF version | ||
| Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| fmptd2f.1 | ⊢ Ⅎ𝑥𝜑 | 
| fmptd2f.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | 
| Ref | Expression | 
|---|---|
| fmptd2f | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fmptd2f.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | fmptd2f.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
| 3 | eqid 2736 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | 1, 2, 3 | fmptdf 7136 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1782 ∈ wcel 2107 ↦ cmpt 5224 ⟶wf 6556 | 
| 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-in 3957 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-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: climinf2mpt 45734 climinfmpt 45735 limsupvaluzmpt 45737 limsupre2mpt 45750 limsupre3mpt 45754 limsupreuzmpt 45759 supcnvlimsupmpt 45761 liminfvalxrmpt 45806 liminflbuz2 45835 dvnprodlem1 45966 sge0z 46395 sge0f1o 46402 smfsupmpt 46835 smfinfmpt 46839 smflimsupmpt 46849 smfliminfmpt 46852 smfsupdmmbllem 46864 smfinfdmmbllem 46868 | 
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