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| Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelcosslem | Structured version Visualization version GIF version | ||
| Description: Lemma for the left side of the refrelcoss3 38464 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.) |
| Ref | Expression |
|---|---|
| refrelcosslem | ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralel 3064 | . 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 | |
| 2 | eldmcoss2 38460 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥)) | |
| 3 | 2 | elv 3485 | . . 3 ⊢ (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥) |
| 4 | 3 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥) |
| 5 | 1, 4 | mpbi 230 | 1 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 class class class wbr 5143 dom cdm 5685 ≀ ccoss 38182 |
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 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-opab 5206 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-coss 38412 |
| This theorem is referenced by: refrelcoss3 38464 eqvrelcoss3 38619 |
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