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Mirrors > Home > MPE Home > Th. List > ecss | Structured version Visualization version GIF version |
Description: An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ecss.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
Ref | Expression |
---|---|
ecss | ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ec 8280 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imassrn 5933 | . . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅 | |
3 | 1, 2 | eqsstri 3998 | . 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅 |
4 | ecss.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
5 | errn 8300 | . . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝑋) |
7 | 3, 6 | sseqtrid 4016 | 1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ⊆ wss 3933 {csn 4557 ran crn 5549 “ cima 5551 Er wer 8275 [cec 8276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-er 8278 df-ec 8280 |
This theorem is referenced by: qsss 8347 divsfval 16808 sylow1lem5 18656 sylow2alem2 18672 sylow2blem1 18674 sylow3lem3 18683 vitalilem2 24137 qsalrel 39003 |
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