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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 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 𝜑) | 
| 3 | 2 | rabeqcda 3448 | . 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴) | 
| 4 | 3 | mptru 1547 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 {crab 3436 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 | 
| This theorem is referenced by: bday0s 27873 2clwwlk2 30367 numclwwlk3lem2lem 30402 fply1 33584 elnanelprv 35434 ipolub0 48881 ipoglb0 48883 | 
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