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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 5147 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵)) | |
| 2 | 1 | spcegv 3597 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥)) |
| 3 | 2 | imp 406 | . . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥) |
| 4 | eldmg 5909 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
| 5 | 3, 4 | imbitrrid 246 | . 2 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)) |
| 6 | 5 | 3impib 1117 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∃wex 1779 ∈ wcel 2108 class class class wbr 5143 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: breldmd 5923 brelrng 5952 releldm 5955 sossfld 6206 brtpos 8260 fprresex 8335 wfrlem17OLD 8365 tfrlem9a 8426 perpln1 28718 lmdvg 33952 esumcvgsum 34089 climeldmeq 45680 climfv 45706 climxlim2 45861 sge0isum 46442 smflimsuplem6 46840 eubrdm 47048 funressneu 47059 tz6.12-afv 47185 rlimdmafv 47189 tz6.12-afv2 47252 rlimdmafv2 47270 |
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