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Mirrors > Home > MPE Home > Th. List > dfrab2 | Structured version Visualization version GIF version |
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.) |
Ref | Expression |
---|---|
dfrab2 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrab3 4325 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
2 | incom 4217 | . 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
3 | 1, 2 | eqtri 2763 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2712 {crab 3433 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 |
This theorem is referenced by: rabdif 4327 dfpred3 6334 predres 6362 lubdm 18409 glbdm 18422 psrbagsn 22105 ismbl 25575 lrrecse 27990 lrrecpred 27992 eulerpartgbij 34354 orvcval4 34442 fvline2 36128 abeqin 38234 nznngen 44312 |
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