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Mirrors > Home > MPE Home > Th. List > Mathboxes > eccnvepres2 | Structured version Visualization version GIF version |
Description: The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.) |
Ref | Expression |
---|---|
eccnvepres2 | ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecres2 38183 | . 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = [𝐵]◡ E ) | |
2 | eccnvep 38186 | . 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]◡ E = 𝐵) | |
3 | 1, 2 | eqtrd 2774 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 E cep 5602 ◡ccnv 5698 ↾ cres 5701 [cec 8757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-eprel 5603 df-xp 5705 df-rel 5706 df-cnv 5707 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-ec 8761 |
This theorem is referenced by: eccnvepres3 38190 |
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