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Mirrors > Home > MPE Home > Th. List > rabeqc | Structured version Visualization version GIF version |
Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.) |
Ref | Expression |
---|---|
rabeqc.1 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Ref | Expression |
---|---|
rabeqc | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqc.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝜑) | |
2 | 1 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 𝜑) |
3 | 2 | rabeqcda 3444 | . 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴) |
4 | 3 | mptru 1549 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 {crab 3433 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 |
This theorem is referenced by: bday0s 27329 2clwwlk2 29601 numclwwlk3lem2lem 29636 fply1 32637 elnanelprv 34420 ipolub0 47617 ipoglb0 47619 |
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