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Mirrors > Home > MPE Home > Th. List > rabss | Structured version Visualization version GIF version |
Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.) |
Ref | Expression |
---|---|
rabss | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3419 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | sseq1i 4005 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵) |
3 | abss 4054 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵)) | |
4 | impexp 449 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
5 | 4 | albii 1813 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) |
6 | df-ral 3051 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
7 | 5, 6 | bitr4i 277 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
8 | 2, 3, 7 | 3bitri 296 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 ∈ wcel 2098 {cab 2702 ∀wral 3050 {crab 3418 ⊆ wss 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rab 3419 df-ss 3961 |
This theorem is referenced by: rabssdv 4068 fnsuppres 8196 wemapso2lem 9577 tskwe2 10798 grothac 10855 uzwo3 12960 fsuppmapnn0fiub0 13994 dvdsssfz1 16298 phibndlem 16742 dfphi2 16746 ramval 16980 mgmidsssn0 18635 istopon 22858 ordtrest2lem 23151 filssufilg 23859 cfinufil 23876 blsscls2 24457 nmhmcn 25091 ovolshftlem2 25483 atansssdm 26910 leftf 27838 rightf 27839 umgrres1lem 29195 upgrres1 29198 sspval 30605 ubthlem2 30753 ordtrest2NEWlem 33654 truae 33993 poimirlem30 37254 nnubfi 37354 prnc 37671 supminfrnmpt 44965 supminfxrrnmpt 44991 itgperiod 45507 fourierdlem81 45713 ovnsupge0 46083 smflimlem2 46298 |
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