Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4243 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
2 | incom 4135 | . 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
3 | 1, 2 | eqtri 2766 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 {cab 2715 {crab 3068 ∩ cin 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-in 3894 |
This theorem is referenced by: dfpred3 6213 predres 6242 lubdm 18069 glbdm 18082 psrbagsn 21271 ismbl 24690 eulerpartgbij 32339 orvcval4 32427 lrrecse 34099 lrrecpred 34101 fvline2 34448 abeqin 36392 rabdif 40184 nznngen 41934 |
Copyright terms: Public domain | W3C validator |