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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 5621 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
5 | 1, 2, 3, 4 | syl3anc 1351 | 1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2048 class class class wbr 4923 dom cdm 5400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-dm 5410 |
This theorem is referenced by: fvelimad 40891 climresdm 41508 xlimliminflimsup 41520 |
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