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Mirrors > Home > MPE Home > Th. List > Mathboxes > elec1cnvxrn2 | Structured version Visualization version GIF version |
Description: Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.) |
Ref | Expression |
---|---|
elec1cnvxrn2 | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6125 | . . 3 ⊢ Rel ◡(𝑅 ⋉ 𝑆) | |
2 | relelec 8791 | . . 3 ⊢ (Rel ◡(𝑅 ⋉ 𝑆) → (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ 𝐴◡(𝑅 ⋉ 𝑆)𝐵)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ 𝐴◡(𝑅 ⋉ 𝑆)𝐵) |
4 | br1cnvxrn2 38378 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴◡(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) | |
5 | 3, 4 | bitrid 283 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ◡ccnv 5688 Rel wrel 5694 [cec 8742 ⋉ cxrn 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-ec 8746 df-xrn 38353 |
This theorem is referenced by: rnxrn 38380 |
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