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Mirrors > Home > MPE Home > Th. List > ercl | Structured version Visualization version 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 8287 | . . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝑅) |
4 | ersym.2 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
5 | releldm 5807 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
7 | erdm 8288 | . . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 6, 8 | eleqtrd 2912 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 Rel wrel 5553 Er wer 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dm 5558 df-er 8278 |
This theorem is referenced by: ercl2 8291 erthi 8329 qliftfun 8371 efgcpbl2 18812 frgpcpbl 18814 prter3 35898 |
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