Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eccnvepres | Structured version Visualization version GIF version |
Description: Restricted converse epsilon coset of 𝐵. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.) |
Ref | Expression |
---|---|
eccnvepres | ⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvep 35407 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡ E 𝑥 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | anbi1cd 633 | . . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥) ↔ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))) |
3 | 2 | abbidv 2882 | . 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)}) |
4 | ecres 35416 | . 2 ⊢ [𝐵](◡ E ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)} | |
5 | df-rab 3144 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)} | |
6 | 3, 4, 5 | 3eqtr4g 2878 | 1 ⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 {crab 3139 class class class wbr 5057 E cep 5457 ◡ccnv 5547 ↾ cres 5550 [cec 8276 |
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-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 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-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8280 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |