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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 8274 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imassrn 5907 | . . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅 | |
3 | 1, 2 | eqsstri 3949 | . 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅 |
4 | ecss.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
5 | errn 8294 | . . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝑋) |
7 | 3, 6 | sseqtrid 3967 | 1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ⊆ wss 3881 {csn 4525 ran crn 5520 “ cima 5522 Er wer 8269 [cec 8270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-er 8272 df-ec 8274 |
This theorem is referenced by: qsss 8341 divsfval 16812 sylow1lem5 18719 sylow2alem2 18735 sylow2blem1 18737 sylow3lem3 18746 vitalilem2 24213 qsalrel 39420 |
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