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Mathbox for Peter Mazsa |
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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 36848 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) | |
2 | 3anass 1095 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) | |
3 | 2 | 2exbii 1851 | . . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) |
4 | 1, 3 | bitrdi 286 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)))) |
5 | elopab 5484 | . . 3 ⊢ (𝑥 ∈ {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))) | |
6 | 4, 5 | bitr4di 288 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝑥 ∈ {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})) |
7 | 6 | eqrdv 2734 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 〈cop 4592 class class class wbr 5105 {copab 5167 [cec 8646 ⋉ cxrn 36633 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
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-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fo 6502 df-fv 6504 df-1st 7921 df-2nd 7922 df-ec 8650 df-xrn 36833 |
This theorem is referenced by: disjecxrn 36851 br1cosscnvxrn 36936 |
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