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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 8754 | . . 3 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
| 2 | relssdmrn 6288 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | 
| 4 | erdm 8755 | . . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
| 5 | errn 8767 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
| 6 | 4, 5 | xpeq12d 5716 | . 2 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴)) | 
| 7 | 3, 6 | sseqtrd 4020 | 1 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ⊆ wss 3951 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 Er wer 8742 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-er 8745 | 
| This theorem is referenced by: erex 8769 riiner 8830 efgval 19735 qtophaus 33835 | 
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