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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 38065 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.) |
Ref | Expression |
---|---|
refrelcosslem | ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralel 3053 | . 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 | |
2 | eldmcoss2 38061 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥)) | |
3 | 2 | elv 3467 | . . 3 ⊢ (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥) |
4 | 3 | ralbii 3082 | . 2 ⊢ (∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥) |
5 | 1, 4 | mpbi 229 | 1 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 class class class wbr 5149 dom cdm 5678 ≀ ccoss 37779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-coss 38013 |
This theorem is referenced by: refrelcoss3 38065 eqvrelcoss3 38220 |
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