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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 3427 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | inab 4294 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | abid2 2865 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
4 | 3 | ineq1i 4203 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
5 | 1, 2, 4 | 3eqtr2i 2760 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 {crab 3426 ∩ cin 3942 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-in 3950 |
This theorem is referenced by: dfrab2 4305 notrab 4306 dfrab3ss 4307 dfif3 4537 dffr3 6091 dfse2 6092 tz6.26OLD 6342 rabfi 9268 dfsup2 9438 ressmplbas2 21920 clsocv 25129 hasheuni 33613 bj-inrab3 36316 bj-reabeq 36415 hashnzfz 43636 |
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