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Mirrors > Home > MPE Home > Th. List > epin | Structured version Visualization version GIF version |
Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) Put in closed form. (Revised by BJ, 16-Oct-2024.) |
Ref | Expression |
---|---|
epin | ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3478 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | eliniseg 6093 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴)) |
3 | epelg 5581 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | bitrd 278 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴)) |
5 | 4 | eqrdv 2730 | 1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4628 class class class wbr 5148 E cep 5579 ◡ccnv 5675 “ cima 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-eprel 5580 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: epini 6095 |
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