Theorem 2idlmex 14514

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	⊢ (𝑈 ∈ 𝑇 → 𝑊 ∈ V)

Proof of Theorem 2idlmex

Step	Hyp	Ref	Expression
1		mptrel 4858	. . . 4 ⊢ Rel (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
2		df-2idl 14513	. . . . 5 ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
3	2	releqi 4809	. . . 4 ⊢ (Rel 2Ideal ↔ Rel (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟)))))
4	1, 3	mpbir 146	. . 3 ⊢ Rel 2Ideal
5		2idlmex.i	. . . . 5 ⊢ 𝑇 = (2Ideal‘𝑊)
6	5	eleq2i 2298	. . . 4 ⊢ (𝑈 ∈ 𝑇 ↔ 𝑈 ∈ (2Ideal‘𝑊))
7	6	biimpi 120	. . 3 ⊢ (𝑈 ∈ 𝑇 → 𝑈 ∈ (2Ideal‘𝑊))
8		relelfvdm 5671	. . 3 ⊢ ((Rel 2Ideal ∧ 𝑈 ∈ (2Ideal‘𝑊)) → 𝑊 ∈ dom 2Ideal)
9	4, 7, 8	sylancr 414	. 2 ⊢ (𝑈 ∈ 𝑇 → 𝑊 ∈ dom 2Ideal)
10	9	elexd 2816	1 ⊢ (𝑈 ∈ 𝑇 → 𝑊 ∈ V)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∩ cin 3199   ↦ cmpt 4150  dom cdm 4725  Rel wrel 4730  ‘cfv 5326  opp_rcoppr 14079  LIdealclidl 14480  2Idealc2idl 14512

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-dm 4735  df-iota 5286  df-fv 5334  df-2idl 14513

This theorem is referenced by:  2idlval  14515  2idlelb  14518  2idllidld  14519  2idlridld  14520  2idlbas  14528