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| Description: Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.) | 
| Ref | Expression | 
|---|---|
| elcnv | ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-cnv 5692 | . . 3 ⊢ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} | |
| 2 | 1 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ ◡𝑅 ↔ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥}) | 
| 3 | elopab 5531 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑦𝑅𝑥)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 {copab 5204 ◡ccnv 5683 | 
| 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-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-cnv 5692 | 
| This theorem is referenced by: elcnv2 5887 gsummpt2co 33052 | 
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