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Mirrors > Home > MPE Home > Th. List > brab1 | Structured version Visualization version 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 | breq1 5069 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝐴 ↔ 𝑦𝑅𝐴)) | |
2 | breq1 5069 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝐴 ↔ 𝑥𝑅𝐴)) | |
3 | 1, 2 | sbcie2g 3811 | . . 3 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴)) |
4 | 3 | elv 3499 | . 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴) |
5 | df-sbc 3773 | . 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) | |
6 | 4, 5 | bitr3i 279 | 1 ⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 {cab 2799 Vcvv 3494 [wsbc 3772 class class class wbr 5066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 |
This theorem is referenced by: (None) |
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