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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 5866 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5106 dom cdm 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-dm 5644 |
This theorem is referenced by: fvelimad 6910 climresdm 44177 xlimliminflimsup 44189 |
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