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Mirrors > Home > MPE Home > Th. List > dfrab3 | Structured version Visualization version GIF version |
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
dfrab3 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3059 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | inab 4038 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | abid2 2883 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
4 | 3 | ineq1i 3953 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
5 | 1, 2, 4 | 3eqtr2i 2788 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1632 ∈ wcel 2139 {cab 2746 {crab 3054 ∩ cin 3714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rab 3059 df-v 3342 df-in 3722 |
This theorem is referenced by: dfrab2 4046 notrab 4047 dfrab3ss 4048 dfif3 4244 dffr3 5656 dfse2 5657 tz6.26 5872 rabfi 8350 dfsup2 8515 ressmplbas2 19657 clsocv 23249 hasheuni 30456 bj-inrab3 33231 hashnzfz 39021 |
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