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 34036 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦))) | |
2 | eldmg 5319 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦)) | |
3 | eldm4 34037 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝑦 ∈ [𝐵]𝑅)) | |
4 | 2, 3 | bitr3d 270 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦 𝐵𝑅𝑦 ↔ ∃𝑦 𝑦 ∈ [𝐵]𝑅)) |
5 | 4 | anbi2d 740 | . 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ ∃𝑦 𝐵𝑅𝑦) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅))) |
6 | 1, 5 | bitrd 268 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ [𝐵]𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 class class class wbr 4653 dom cdm 5114 ↾ cres 5116 [cec 7740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ec 7744 |
This theorem is referenced by: eldmqsres 34051 |
Copyright terms: Public domain | W3C validator |