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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ecxrncnvep | Structured version Visualization version GIF version | ||
| Description: The (𝑅 ⋉ ◡ E )-coset of a set. (Contributed by Peter Mazsa, 22-May-2021.) |
| Ref | Expression |
|---|---|
| ecxrncnvep | ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ ◡ E ) = {〈𝑦, 𝑧〉 ∣ (𝑧 ∈ 𝐴 ∧ 𝐴𝑅𝑦)}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ecxrn 38858 | . 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ ◡ E ) = {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴◡ E 𝑧)}) | |
| 2 | brcnvep 38722 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝑧 ↔ 𝑧 ∈ 𝐴)) | |
| 3 | 2 | anbi1cd 644 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴𝑅𝑦 ∧ 𝐴◡ E 𝑧) ↔ (𝑧 ∈ 𝐴 ∧ 𝐴𝑅𝑦))) |
| 4 | 3 | opabbidv 5165 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴◡ E 𝑧)} = {〈𝑦, 𝑧〉 ∣ (𝑧 ∈ 𝐴 ∧ 𝐴𝑅𝑦)}) |
| 5 | 1, 4 | eqtrd 2796 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ ◡ E ) = {〈𝑦, 𝑧〉 ∣ (𝑧 ∈ 𝐴 ∧ 𝐴𝑅𝑦)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 class class class wbr 5099 {copab 5161 E cep 5544 ◡ccnv 5644 [cec 8669 ⋉ cxrn 38626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389 ax-un 7712 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-eprel 5545 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-fo 6521 df-fv 6523 df-1st 7964 df-2nd 7965 df-ec 8673 df-xrn 38832 |
| This theorem is referenced by: (None) |
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