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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmres | Structured version Visualization version GIF version |
Description: Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 9-Jan-2019.) |
Ref | Expression |
---|---|
eldmres | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmg 5891 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ ∃𝑦 𝐵(𝑅 ↾ 𝐴)𝑦)) | |
2 | brres 5981 | . . . . 5 ⊢ (𝑦 ∈ V → (𝐵(𝑅 ↾ 𝐴)𝑦 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦))) | |
3 | 2 | elv 3474 | . . . 4 ⊢ (𝐵(𝑅 ↾ 𝐴)𝑦 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)) |
4 | 3 | exbii 1842 | . . 3 ⊢ (∃𝑦 𝐵(𝑅 ↾ 𝐴)𝑦 ↔ ∃𝑦(𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)) |
5 | 19.42v 1949 | . . 3 ⊢ (∃𝑦(𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)) | |
6 | 4, 5 | bitri 275 | . 2 ⊢ (∃𝑦 𝐵(𝑅 ↾ 𝐴)𝑦 ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦)) |
7 | 1, 6 | bitrdi 287 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 dom cdm 5669 ↾ cres 5671 |
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-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-dm 5679 df-res 5681 |
This theorem is referenced by: eldmressnALTV 37652 eldmres2 37655 |
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