Theorem 2idlmex 14838

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 e. T -> W e. _V )

Proof of Theorem 2idlmex

Step	Hyp	Ref	Expression
1		mptrel 4908	. . . 4 |- Rel ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
2		df-2idl 14837	. . . . 5 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3	2	releqi 4858	. . . 4 |- ( Rel 2Ideal <-> Rel ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) ) )
4	1, 3	mpbir 146	. . 3 |- Rel 2Ideal
5		2idlmex.i	. . . . 5 |- T = (2Ideal ` W )
6	5	eleq2i 2305	. . . 4 |- ( U e. T <-> U e. (2Ideal ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. T -> U e. (2Ideal ` W ) )
8		relelfvdm 5727	. . 3 |- ( ( Rel 2Ideal /\ U e. (2Ideal ` W ) ) -> W e. dom 2Ideal )
9	4, 7, 8	sylancr 418	. 2 |- ( U e. T -> W e. dom 2Ideal )
10	9	elexd 2835	1 |- ( U e. T -> W e. _V )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   |-> cmpt 4192   dom cdm 4774   Rel wrel 4779   ` cfv 5377  opp_rcoppr 14372  LIdealclidl 14804  2Idealc2idl 14836

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-xp 4780  df-rel 4781  df-dm 4784  df-iota 5337  df-fv 5385  df-2idl 14837

This theorem is used by:  2idlval  14839  2idlelb  14842  2idllidld  14843  2idlridld  14844  2idlbas  14852