![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ecres | Structured version Visualization version GIF version |
Description: Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.) |
Ref | Expression |
---|---|
ecres | ⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elecres 34539 | . . 3 ⊢ (𝑥 ∈ V → (𝑥 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥))) | |
2 | 1 | elv 3389 | . 2 ⊢ (𝑥 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)) |
3 | 2 | abbi2i 2915 | 1 ⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {cab 2785 Vcvv 3385 class class class wbr 4843 ↾ cres 5314 [cec 7980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ec 7984 |
This theorem is referenced by: eccnvepres 34542 |
Copyright terms: Public domain | W3C validator |