Theorem 2idlmex 14378

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 4824	. . . 4 |- Rel ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
2		df-2idl 14377	. . . . 5 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3	2	releqi 4776	. . . 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 2274	. . . 4 |- ( U e. T <-> U e. (2Ideal ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. T -> U e. (2Ideal ` W ) )
8		relelfvdm 5631	. . 3 |- ( ( Rel 2Ideal /\ U e. (2Ideal ` W ) ) -> W e. dom 2Ideal )
9	4, 7, 8	sylancr 414	. 2 |- ( U e. T -> W e. dom 2Ideal )
10	9	elexd 2790	1 |- ( U e. T -> W e. _V )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   |-> cmpt 4121   dom cdm 4693   Rel wrel 4698   ` cfv 5290  opp_rcoppr 13944  LIdealclidl 14344  2Idealc2idl 14376

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 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-xp 4699  df-rel 4700  df-dm 4703  df-iota 5251  df-fv 5298  df-2idl 14377

This theorem is referenced by:  2idlval  14379  2idlelb  14382  2idllidld  14383  2idlridld  14384  2idlbas  14392