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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvepresex | Structured version Visualization version GIF version | ||
| Description: Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.) | 
| Ref | Expression | 
|---|---|
| cnvepresex | ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cnvepres 38299 | . 2 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
| 2 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
| 3 | abid2 2879 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥 | |
| 4 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 5 | 3, 4 | eqeltri 2837 | . . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V | 
| 6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V) | 
| 7 | 2, 6 | opabex3d 7990 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V) | 
| 8 | 1, 7 | eqeltrid 2845 | 1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 {cab 2714 Vcvv 3480 {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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 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-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 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: eccnvepex 38336 cnvepimaex 38337 cnvepima 38338 xrncnvepresex 38409 1cosscnvepresex 38422 cnvepresdmqss 38653 eleldisjseldisj 38730 mpets2 38842 | 
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