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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 epsilon 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 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eliniseg 5413 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴) |
5 | 1 | epelc 4951 | . . 3 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴) |
6 | 4, 5 | bitri 263 | . 2 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴) |
7 | 6 | eqriv 2607 | 1 ⊢ (◡ E “ {𝐴}) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 class class class wbr 4583 E cep 4947 ◡ccnv 5037 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: infxpenlem 8719 fz1isolem 13102 |
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