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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldm1cossres | Structured version Visualization version GIF version |
Description: Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.) |
Ref | Expression |
---|---|
eldm1cossres | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑢𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmcoss 37784 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 𝑢(𝑅 ↾ 𝐴)𝐵)) | |
2 | brres 5978 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑢(𝑅 ↾ 𝐴)𝐵 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵))) | |
3 | 2 | exbidv 1916 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑢 𝑢(𝑅 ↾ 𝐴)𝐵 ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵))) |
4 | 1, 3 | bitrd 279 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵))) |
5 | df-rex 3063 | . 2 ⊢ (∃𝑢 ∈ 𝐴 𝑢𝑅𝐵 ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵)) | |
6 | 4, 5 | bitr4di 289 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑢𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ∃wrex 3062 class class class wbr 5138 dom cdm 5666 ↾ cres 5668 ≀ ccoss 37499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-coss 37737 |
This theorem is referenced by: eldm1cossres2 37787 |
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