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Mirrors > Home > ILE Home > Th. List > ercl | GIF version |
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersym.1 | ⊢ (φ → 𝑅 Er 𝑋) |
ersym.2 | ⊢ (φ → A𝑅B) |
Ref | Expression |
---|---|
ercl | ⊢ (φ → A ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersym.1 | . . . 4 ⊢ (φ → 𝑅 Er 𝑋) | |
2 | errel 6051 | . . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (φ → Rel 𝑅) |
4 | ersym.2 | . . 3 ⊢ (φ → A𝑅B) | |
5 | releldm 4512 | . . 3 ⊢ ((Rel 𝑅 ∧ A𝑅B) → A ∈ dom 𝑅) | |
6 | 3, 4, 5 | syl2anc 391 | . 2 ⊢ (φ → A ∈ dom 𝑅) |
7 | erdm 6052 | . . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 1, 7 | syl 14 | . 2 ⊢ (φ → dom 𝑅 = 𝑋) |
9 | 6, 8 | eleqtrd 2113 | 1 ⊢ (φ → A ∈ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 class class class wbr 3755 dom cdm 4288 Rel wrel 4293 Er wer 6039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-dm 4298 df-er 6042 |
This theorem is referenced by: ercl2 6055 erthi 6088 qliftfun 6124 |
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