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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 3430 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | inab 4300 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | abid2 2867 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
4 | 3 | ineq1i 4208 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∩ {𝑥 ∣ 𝜑}) = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
5 | 1, 2, 4 | 3eqtr2i 2762 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 {crab 3429 ∩ cin 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-in 3954 |
This theorem is referenced by: dfrab2 4311 notrab 4312 dfrab3ss 4313 dfif3 4543 dffr3 6103 dfse2 6104 tz6.26OLD 6354 rabfi 9294 dfsup2 9468 ressmplbas2 21965 clsocv 25191 hasheuni 33704 bj-inrab3 36407 bj-reabeq 36506 hashnzfz 43757 |
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