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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 36398 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 5036 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝑆 ↾ 𝐶)𝐵) |
3 | brres 5832 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑆 ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) | |
4 | 2, 3 | syl5bb 286 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 ↾ cres 5527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-xp 5531 df-res 5537 |
This theorem is referenced by: brers 36402 |
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