Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3147 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | sseq1i 3995 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵) |
3 | abss 4040 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵)) | |
4 | impexp 453 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
5 | 4 | albii 1820 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) |
6 | df-ral 3143 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
7 | 5, 6 | bitr4i 280 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
8 | 2, 3, 7 | 3bitri 299 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∈ wcel 2114 {cab 2799 ∀wral 3138 {crab 3142 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-in 3943 df-ss 3952 |
This theorem is referenced by: rabssdv 4051 fnsuppres 7857 wemapso2lem 9016 tskwe2 10195 grothac 10252 uzwo3 12344 fsuppmapnn0fiub0 13362 dvdsssfz1 15668 phibndlem 16107 dfphi2 16111 ramval 16344 mgmidsssn0 17882 istopon 21520 ordtrest2lem 21811 filssufilg 22519 cfinufil 22536 blsscls2 23114 nmhmcn 23724 ovolshftlem2 24111 atansssdm 25511 umgrres1lem 27092 upgrres1 27095 sspval 28500 ubthlem2 28648 ordtrest2NEWlem 31165 truae 31502 poimirlem30 34937 nnubfi 35040 prnc 35360 supminfrnmpt 41739 supminfxrrnmpt 41767 itgperiod 42286 fourierdlem81 42492 ovnsupge0 42859 smflimlem2 43068 |
Copyright terms: Public domain | W3C validator |