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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 6121 | . . 3 ⊢ Rel ◡(𝑅 ⋉ 𝑆) | |
| 2 | relelec 8793 | . . 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 38398 | . 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 1539 ∃wex 1778 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 ◡ccnv 5683 Rel wrel 5689 [cec 8744 ⋉ cxrn 38182 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-1st 8015 df-2nd 8016 df-ec 8748 df-xrn 38373 | 
| This theorem is referenced by: rnxrn 38400 | 
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