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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 6537 | . . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → Rel 𝑅) |
4 | ersym.2 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
5 | releldm 4857 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
6 | 3, 4, 5 | syl2anc 411 | . 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
7 | erdm 6538 | . . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 1, 7 | syl 14 | . 2 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 6, 8 | eleqtrd 2256 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 dom cdm 4622 Rel wrel 4627 Er wer 6525 |
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 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4628 df-rel 4629 df-dm 4632 df-er 6528 |
This theorem is referenced by: ercl2 6541 erthi 6574 qliftfun 6610 |
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