Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5778 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5053 dom cdm 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-dm 5561 |
This theorem is referenced by: fvelimad 6779 climresdm 43066 xlimliminflimsup 43078 |
Copyright terms: Public domain | W3C validator |