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Theorem 2idlmex 14338
Description: Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
Hypothesis
Ref Expression
2idlmex.i 𝑇 = (2Ideal‘𝑊)
Assertion
Ref Expression
2idlmex (𝑈𝑇𝑊 ∈ V)

Proof of Theorem 2idlmex
StepHypRef Expression
1 mptrel 4814 . . . 4 Rel (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
2 df-2idl 14337 . . . . 5 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
32releqi 4766 . . . 4 (Rel 2Ideal ↔ Rel (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟)))))
41, 3mpbir 146 . . 3 Rel 2Ideal
5 2idlmex.i . . . . 5 𝑇 = (2Ideal‘𝑊)
65eleq2i 2273 . . . 4 (𝑈𝑇𝑈 ∈ (2Ideal‘𝑊))
76biimpi 120 . . 3 (𝑈𝑇𝑈 ∈ (2Ideal‘𝑊))
8 relelfvdm 5621 . . 3 ((Rel 2Ideal ∧ 𝑈 ∈ (2Ideal‘𝑊)) → 𝑊 ∈ dom 2Ideal)
94, 7, 8sylancr 414 . 2 (𝑈𝑇𝑊 ∈ dom 2Ideal)
109elexd 2787 1 (𝑈𝑇𝑊 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1373  wcel 2177  Vcvv 2773  cin 3169  cmpt 4113  dom cdm 4683  Rel wrel 4688  cfv 5280  opprcoppr 13904  LIdealclidl 14304  2Idealc2idl 14336
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-xp 4689  df-rel 4690  df-dm 4693  df-iota 5241  df-fv 5288  df-2idl 14337
This theorem is referenced by:  2idlval  14339  2idlelb  14342  2idllidld  14343  2idlridld  14344  2idlbas  14352
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