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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 35705 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.) |
Ref | Expression |
---|---|
refrelcosslem | ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralel 3151 | . 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 | |
2 | eldmcoss2 35701 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥)) | |
3 | 2 | elv 3501 | . . 3 ⊢ (𝑥 ∈ dom ≀ 𝑅 ↔ 𝑥 ≀ 𝑅𝑥) |
4 | 3 | ralbii 3167 | . 2 ⊢ (∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥) |
5 | 1, 4 | mpbi 232 | 1 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 class class class wbr 5068 dom cdm 5557 ≀ ccoss 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-coss 35661 |
This theorem is referenced by: refrelcoss3 35705 eqvrelcoss3 35855 |
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