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Mirrors > Home > ILE Home > Th. List > dfec2 | GIF version |
Description: Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
dfec2 | ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ec 6534 | . 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
2 | imasng 4992 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) | |
3 | 1, 2 | eqtrid 2222 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 {cab 2163 {csn 3592 class class class wbr 4002 “ cima 4628 [cec 6530 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-ec 6534 |
This theorem is referenced by: eqglact 13015 |
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