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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 35549 | . 2 ⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | |
2 | elex 3513 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | abid2 2957 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} = 𝑥 | |
4 | vex 3498 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | eqeltri 2909 | . . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ 𝑥} ∈ V) |
7 | 2, 6 | opabex3d 7660 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} ∈ V) |
8 | 1, 7 | eqeltrid 2917 | 1 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 {cab 2799 Vcvv 3495 {copab 5121 E cep 5459 ◡ccnv 5549 ↾ cres 5552 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-eprel 5460 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 |
This theorem is referenced by: eccnvepex 35586 cnvepimaex 35587 cnvepima 35588 xrncnvepresex 35650 1cosscnvepresex 35660 cnvepresdmqss 35880 eleldisjseldisj 35956 |
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