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Mirrors > Home > MPE Home > Th. List > epini | 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.) |
Ref | Expression |
---|---|
epini.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
epini | ⊢ (◡ E “ {𝐴}) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epini.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | vex 3402 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eliniseg 5943 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴) |
5 | 1 | epeli 5447 | . . 3 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴) |
6 | 4, 5 | bitri 278 | . 2 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴) |
7 | 6 | eqriv 2733 | 1 ⊢ (◡ E “ {𝐴}) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 Vcvv 3398 {csn 4527 class class class wbr 5039 E cep 5444 ◡ccnv 5535 “ cima 5539 |
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 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-eprel 5445 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 |
This theorem is referenced by: infxpenlem 9592 fz1isolem 13992 |
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