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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 5884 | . 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
3 | dmmptssf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | ssrab2f 40229 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
5 | 2, 4 | eqsstri 3854 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 Ⅎwnfc 2919 {crab 3094 Vcvv 3398 ⊆ wss 3792 ↦ cmpt 4965 dom cdm 5355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-mpt 4966 df-xp 5361 df-rel 5362 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 |
This theorem is referenced by: limsupequzmpt2 40858 liminfequzmpt2 40931 |
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