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 4198 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
2 | incom 4091 | . 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
3 | 1, 2 | eqtri 2761 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cab 2716 {crab 3057 ∩ cin 3842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-in 3850 |
This theorem is referenced by: dfpred3 6139 lubdm 17707 glbdm 17720 psrbagsn 20877 ismbl 24280 eulerpartgbij 31911 orvcval4 31999 lrrecse 33744 lrrecpred 33746 fvline2 34093 abeqin 36037 rabdif 39797 nznngen 41494 |
Copyright terms: Public domain | W3C validator |