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Mathbox for Peter Mazsa |
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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 37336 or df-parts 37438) 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 5147 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝑆 ↾ 𝐶)𝐵) |
3 | brres 5980 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑆 ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) | |
4 | 2, 3 | bitrid 282 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ↾ cres 5671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-res 5681 |
This theorem is referenced by: brers 37340 brparts 37444 |
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