![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elecres | Structured version Visualization version GIF version |
Description: Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.) |
Ref | Expression |
---|---|
elecres | ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 6011 | . . 3 ⊢ Rel (𝑅 ↾ 𝐴) | |
2 | relelec 8771 | . . 3 ⊢ (Rel (𝑅 ↾ 𝐴) → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵(𝑅 ↾ 𝐴)𝐶)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵(𝑅 ↾ 𝐴)𝐶) |
4 | brres 5992 | . 2 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | |
5 | 3, 4 | bitrid 282 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5149 ↾ cres 5680 Rel wrel 5683 [cec 8723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 |
This theorem is referenced by: ecres 37880 ecres2 37881 |
Copyright terms: Public domain | W3C validator |