Theorem 2idlmex 14810

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 4903	. . . 4 |- Rel ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
2		df-2idl 14809	. . . . 5 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3	2	releqi 4853	. . . 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 5722	. . 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 )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   |-> cmpt 4187   dom cdm 4769   Rel wrel 4774   ` cfv 5372  opp_rcoppr 14345  LIdealclidl 14776  2Idealc2idl 14808

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 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 4244  ax-pow 4306  ax-pr 4341

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-xp 4775  df-rel 4776  df-dm 4779  df-iota 5332  df-fv 5380  df-2idl 14809

This theorem is referenced by:  2idlval  14811  2idlelb  14814  2idllidld  14815  2idlridld  14816  2idlbas  14824