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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dmmptssf | Structured version Visualization version GIF version | ||
| Description: The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| dmmptssf.1 | ⊢ Ⅎ𝑥𝐴 |
| dmmptssf.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Ref | Expression |
|---|---|
| dmmptssf | ⊢ dom 𝐹 ⊆ 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmmptssf.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | 1 | dmmpt 6260 | . 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
| 3 | dmmptssf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 3 | ssrab2f 45122 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
| 5 | 2, 4 | eqsstri 4030 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 {crab 3436 Vcvv 3480 ⊆ wss 3951 ↦ cmpt 5225 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: rn1st 45280 limsupequzmpt2 45733 liminfequzmpt2 45806 |
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