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Mirrors > Home > ILE Home > Th. List > ecss | 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 6561 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imassrn 4999 | . . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅 | |
3 | 1, 2 | eqsstri 3202 | . 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅 |
4 | ecss.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
5 | errn 6581 | . . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝑋) |
7 | 3, 6 | sseqtrid 3220 | 1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ⊆ wss 3144 {csn 3607 ran crn 4645 “ cima 4647 Er wer 6556 [cec 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-er 6559 df-ec 6561 |
This theorem is referenced by: qsss 6620 divsfvalg 12805 |
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