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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 5043 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
2 | 1 | spcegv 3502 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥)) |
3 | 2 | imp 410 | . . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥) |
4 | eldmg 5752 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
5 | 3, 4 | syl5ibr 249 | . 2 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)) |
6 | 5 | 3impib 1118 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∃wex 1787 ∈ wcel 2112 class class class wbr 5039 dom cdm 5536 |
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 2018 ax-8 2114 ax-9 2122 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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-dm 5546 |
This theorem is referenced by: breldmd 5766 brelrng 5795 releldm 5798 sossfld 6029 brtpos 7955 wfrlem17 8049 tfrlem9a 8100 perpln1 26755 lmdvg 31571 esumcvgsum 31722 climeldmeq 42824 climfv 42850 climxlim2 43005 sge0isum 43583 smflimsuplem6 43973 eubrdm 44145 funressneu 44156 tz6.12-afv 44280 rlimdmafv 44284 tz6.12-afv2 44347 rlimdmafv2 44365 |
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