Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqres | Structured version Visualization version GIF version |
Description: Converting a class constant definition by restriction (like df-ers 35777 or ~? df-parts ) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.) |
Ref | Expression |
---|---|
eqres.1 | ⊢ 𝑅 = (𝑆 ↾ 𝐶) |
Ref | Expression |
---|---|
eqres | ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqres.1 | . . 3 ⊢ 𝑅 = (𝑆 ↾ 𝐶) | |
2 | 1 | breqi 5063 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝑆 ↾ 𝐶)𝐵) |
3 | brres 5853 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑆 ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) | |
4 | 2, 3 | syl5bb 284 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-res 5560 |
This theorem is referenced by: brers 35781 |
Copyright terms: Public domain | W3C validator |