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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 38444 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.) |
Ref | Expression |
---|---|
refrelcosslem | ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralel 3061 | . 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 | |
2 | eldmcoss2 38440 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥)) | |
3 | 2 | elv 3482 | . . 3 ⊢ (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥) |
4 | 3 | ralbii 3090 | . 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 2105 ∀wral 3058 Vcvv 3477 class class class wbr 5147 dom cdm 5688 ≀ ccoss 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-coss 38392 |
This theorem is referenced by: refrelcoss3 38444 eqvrelcoss3 38599 |
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