![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8089 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imassrn 5778 | . . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅 | |
3 | 1, 2 | eqsstri 3884 | . 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅 |
4 | ecss.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
5 | errn 8109 | . . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝑋) |
7 | 3, 6 | syl5sseq 3902 | 1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ⊆ wss 3822 {csn 4435 ran crn 5404 “ cima 5406 Er wer 8084 [cec 8085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-xp 5409 df-rel 5410 df-cnv 5411 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-er 8087 df-ec 8089 |
This theorem is referenced by: qsss 8156 divsfval 16674 sylow1lem5 18500 sylow2alem2 18516 sylow2blem1 18518 sylow3lem3 18527 vitalilem2 23928 qsalrel 38608 |
Copyright terms: Public domain | W3C validator |