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Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelcoss3 | Structured version Visualization version GIF version |
Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 37024. (Contributed by Peter Mazsa, 30-Jul-2019.) |
Ref | Expression |
---|---|
refrelcoss3 | ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refrelcosslem 36970 | . . . 4 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 | |
2 | idinxpssinxp4 36827 | . . . 4 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥) | |
3 | 1, 2 | mpbir 230 | . . 3 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) |
4 | rncossdmcoss 36963 | . . . . 5 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅 | |
5 | 4 | raleqi 3310 | . . . 4 ⊢ (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)) |
6 | 5 | ralbii 3093 | . . 3 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)) |
7 | 3, 6 | mpbir 230 | . 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) |
8 | relcoss 36931 | . 2 ⊢ Rel ≀ 𝑅 | |
9 | 7, 8 | pm3.2i 472 | 1 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wral 3061 class class class wbr 5106 dom cdm 5634 ran crn 5635 Rel wrel 5639 ≀ ccoss 36680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-coss 36919 |
This theorem is referenced by: refrelcoss2 36972 |
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