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Mirrors > Home > MPE Home > Th. List > breldmd | Structured version Visualization version GIF version |
Description: Membership of first of a binary relation in a domain. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
breldmd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
breldmd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
breldmd.3 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
Ref | Expression |
---|---|
breldmd | ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breldmd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | breldmd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
3 | breldmd.3 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
4 | breldmg 5771 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
5 | 1, 2, 3, 4 | syl3anc 1363 | 1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-dm 5558 |
This theorem is referenced by: fvelimad 6725 climresdm 42007 xlimliminflimsup 42019 |
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