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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 3070 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | sseq1i 3929 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵) |
3 | abss 3974 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵)) | |
4 | impexp 454 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
5 | 4 | albii 1827 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) |
6 | df-ral 3066 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
7 | 5, 6 | bitr4i 281 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
8 | 2, 3, 7 | 3bitri 300 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ∈ wcel 2110 {cab 2714 ∀wral 3061 {crab 3065 ⊆ wss 3866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rab 3070 df-v 3410 df-in 3873 df-ss 3883 |
This theorem is referenced by: rabssdv 3988 fnsuppres 7933 wemapso2lem 9168 tskwe2 10387 grothac 10444 uzwo3 12539 fsuppmapnn0fiub0 13566 dvdsssfz1 15879 phibndlem 16323 dfphi2 16327 ramval 16561 mgmidsssn0 18144 istopon 21809 ordtrest2lem 22100 filssufilg 22808 cfinufil 22825 blsscls2 23402 nmhmcn 24017 ovolshftlem2 24407 atansssdm 25816 umgrres1lem 27398 upgrres1 27401 sspval 28804 ubthlem2 28952 ordtrest2NEWlem 31586 truae 31923 leftf 33786 rightf 33787 poimirlem30 35544 nnubfi 35645 prnc 35962 supminfrnmpt 42658 supminfxrrnmpt 42686 itgperiod 43197 fourierdlem81 43403 ovnsupge0 43770 smflimlem2 43979 |
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