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Mirrors > Home > MPE Home > Th. List > breldmg | Structured version Visualization version GIF version |
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.) |
Ref | Expression |
---|---|
breldmg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4587 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
2 | 1 | spcegv 3267 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥)) |
3 | 2 | imp 444 | . . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥) |
4 | eldmg 5241 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
5 | 3, 4 | syl5ibr 235 | . 2 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)) |
6 | 5 | 3impib 1254 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∃wex 1695 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: brelrng 5276 releldm 5279 sossfld 5499 brtpos 7248 wfrlem17 7318 tfrlem9a 7369 perpln1 25405 lmdvg 29327 esumcvgsum 29477 climeldmeq 38732 sge0isum 39320 tz6.12-afv 39902 rlimdmafv 39906 |
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