![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inrab3 | Structured version Visualization version GIF version |
Description: Generalization of dfrab3ss 4304, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.) |
Ref | Expression |
---|---|
bj-inrab3 | ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrab3 4301 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = (𝐵 ∩ {𝑥 ∣ 𝜑}) | |
2 | 1 | ineq2i 4201 | . 2 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑})) |
3 | dfrab3 4301 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
4 | 3 | ineq2i 4201 | . . 3 ⊢ (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐵 ∩ (𝐴 ∩ {𝑥 ∣ 𝜑})) |
5 | incom 4193 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
6 | in12 4212 | . . 3 ⊢ (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑})) = (𝐵 ∩ (𝐴 ∩ {𝑥 ∣ 𝜑})) | |
7 | 4, 5, 6 | 3eqtr4i 2762 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑})) |
8 | 2, 7 | eqtr4i 2755 | 1 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cab 2701 {crab 3424 ∩ cin 3939 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-in 3947 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |