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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvepres | Structured version Visualization version GIF version | ||
| Description: Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.) | 
| Ref | Expression | 
|---|---|
| cnvepres | ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfres2 6059 | . 2 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)} | |
| 2 | brcnvep 38266 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)) | |
| 3 | 2 | elv 3485 | . . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥) | 
| 4 | 3 | anbi2i 623 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | 
| 5 | 4 | opabbii 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥◡ E 𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | 
| 6 | 1, 5 | eqtri 2765 | 1 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 E cep 5583 ◡ccnv 5684 ↾ cres 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-res 5697 | 
| This theorem is referenced by: rncnvepres 38304 n0el2 38334 cnvepresex 38335 | 
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