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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 8281 | . . . 4 ⊢ (𝑅 Er 𝑋 → Rel 𝑅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝑅) |
4 | ersym.2 | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
5 | releldm 5778 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
6 | 3, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
7 | erdm 8282 | . . 3 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 6, 8 | eleqtrd 2892 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 dom cdm 5519 Rel wrel 5524 Er wer 8269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-er 8272 |
This theorem is referenced by: ercl2 8285 erthi 8323 qliftfun 8365 efgcpbl2 18875 frgpcpbl 18877 prter3 36178 |
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