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Mirrors > Home > ILE Home > Th. List > rrgmex | GIF version |
Description: A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.) |
Ref | Expression |
---|---|
rrgmex.e | ⊢ 𝐸 = (RLReg‘𝑅) |
Ref | Expression |
---|---|
rrgmex | ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4784 | . . . 4 ⊢ Rel (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))}) | |
2 | df-rlreg 13738 | . . . . 5 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))}) | |
3 | 2 | releqi 4738 | . . . 4 ⊢ (Rel RLReg ↔ Rel (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})) |
4 | 1, 3 | mpbir 146 | . . 3 ⊢ Rel RLReg |
5 | rrgmex.e | . . . . 5 ⊢ 𝐸 = (RLReg‘𝑅) | |
6 | 5 | eleq2i 2260 | . . . 4 ⊢ (𝐴 ∈ 𝐸 ↔ 𝐴 ∈ (RLReg‘𝑅)) |
7 | 6 | biimpi 120 | . . 3 ⊢ (𝐴 ∈ 𝐸 → 𝐴 ∈ (RLReg‘𝑅)) |
8 | relelfvdm 5578 | . . 3 ⊢ ((Rel RLReg ∧ 𝐴 ∈ (RLReg‘𝑅)) → 𝑅 ∈ dom RLReg) | |
9 | 4, 7, 8 | sylancr 414 | . 2 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ dom RLReg) |
10 | 9 | elexd 2773 | 1 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 ∀wral 2472 {crab 2476 Vcvv 2760 ↦ cmpt 4090 dom cdm 4655 Rel wrel 4660 ‘cfv 5246 (class class class)co 5910 Basecbs 12608 .rcmulr 12686 0gc0g 12857 RLRegcrlreg 13735 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-xp 4661 df-rel 4662 df-dm 4665 df-iota 5207 df-fv 5254 df-rlreg 13738 |
This theorem is referenced by: rrgval 13742 |
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