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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 3405 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | eliniseg 5951 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴)) |
3 | epelg 5450 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | bitrd 282 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴)) |
5 | 4 | eqrdv 2732 | 1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {csn 4531 class class class wbr 5043 E cep 5448 ◡ccnv 5539 “ cima 5543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-eprel 5449 df-xp 5546 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 |
This theorem is referenced by: epini 5953 |
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