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Mirrors > Home > ILE Home > Th. List > brab1 | GIF version |
Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.) |
Ref | Expression |
---|---|
brab1 | ⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . 3 ⊢ 𝑥 ∈ V | |
2 | breq1 3767 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝐴 ↔ 𝑦𝑅𝐴)) | |
3 | breq1 3767 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝐴 ↔ 𝑥𝑅𝐴)) | |
4 | 2, 3 | sbcie2g 2796 | . . 3 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴)) |
5 | 1, 4 | ax-mp 7 | . 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴) |
6 | df-sbc 2765 | . 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) | |
7 | 5, 6 | bitr3i 175 | 1 ⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∈ wcel 1393 {cab 2026 Vcvv 2557 [wsbc 2764 class class class wbr 3764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 |
This theorem is referenced by: (None) |
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