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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eccnvepex | Structured version Visualization version GIF version | ||
| Description: The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.) |
| Ref | Expression |
|---|---|
| eccnvepex | ⊢ [𝐴]◡ E ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snex 5436 | . 2 ⊢ {𝐴} ∈ V | |
| 2 | cnvepresex 38335 | . 2 ⊢ ({𝐴} ∈ V → (◡ E ↾ {𝐴}) ∈ V) | |
| 3 | ecexALTV 38332 | . 2 ⊢ ((◡ E ↾ {𝐴}) ∈ V → [𝐴]◡ E ∈ V) | |
| 4 | 1, 2, 3 | mp2b 10 | 1 ⊢ [𝐴]◡ E ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 {csn 4626 E cep 5583 ◡ccnv 5684 ↾ cres 5687 [cec 8743 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 |
| This theorem is referenced by: (None) |
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