Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ecxrn | Structured version Visualization version GIF version |
Description: The (𝑅 ⋉ 𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
Ref | Expression |
---|---|
ecxrn | ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elecxrn 35653 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) | |
2 | 3anass 1091 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) | |
3 | 2 | 2exbii 1849 | . . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) |
4 | 1, 3 | syl6bb 289 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)))) |
5 | elopab 5414 | . . 3 ⊢ (𝑥 ∈ {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) | |
6 | 4, 5 | syl6bbr 291 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝑥 ∈ {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})) |
7 | 6 | eqrdv 2819 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 {copab 5128 [cec 8287 ⋉ cxrn 35467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-1st 7689 df-2nd 7690 df-ec 8291 df-xrn 35638 |
This theorem is referenced by: br1cosscnvxrn 35729 |
Copyright terms: Public domain | W3C validator |