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Mirrors > Home > MPE Home > Th. List > erssxp | Structured version Visualization version GIF version |
Description: An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
erssxp | ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | errel 8753 | . . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
2 | relssdmrn 6290 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
4 | erdm 8754 | . . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
5 | errn 8766 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
6 | 4, 5 | xpeq12d 5720 | . 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴)) |
7 | 3, 6 | sseqtrd 4036 | 1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3963 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 Er wer 8741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-er 8744 |
This theorem is referenced by: erex 8768 riiner 8829 efgval 19750 qtophaus 33797 |
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