![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmres2 | Structured version Visualization version GIF version |
Description: Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 21-Aug-2020.) |
Ref | Expression |
---|---|
eldmres2 | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmres 38252 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))) | |
2 | eldmg 5912 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦)) | |
3 | eldm4 38256 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐵]𝑅)) | |
4 | 2, 3 | bitr3d 281 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦 𝐵𝑅𝑦 ↔ ∃𝑦 𝑦 ∈ [𝐵]𝑅)) |
5 | 4 | anbi2d 630 | . 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅))) |
6 | 1, 5 | bitrd 279 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 |
This theorem is referenced by: eldmqsres 38269 |
Copyright terms: Public domain | W3C validator |