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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 6439 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imassrn 4900 | . . 3 ⊢ (𝑅 “ {𝐴}) ⊆ ran 𝑅 | |
3 | 1, 2 | eqsstri 3134 | . 2 ⊢ [𝐴]𝑅 ⊆ ran 𝑅 |
4 | ecss.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
5 | errn 6459 | . . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝑋) |
7 | 3, 6 | sseqtrid 3152 | 1 ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ⊆ wss 3076 {csn 3532 ran crn 4548 “ cima 4550 Er wer 6434 [cec 6435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-er 6437 df-ec 6439 |
This theorem is referenced by: qsss 6496 |
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