Theorem 2idlmex 14775

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 4888	. . . 4 |- Rel ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
2		df-2idl 14774	. . . . 5 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3	2	releqi 4838	. . . 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 2301	. . . 4 |- ( U e. T <-> U e. (2Ideal ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. T -> U e. (2Ideal ` W ) )
8		relelfvdm 5707	. . 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 2829	1 |- ( U e. T -> W e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   i^i cin 3213   |-> cmpt 4176   dom cdm 4754   Rel wrel 4759   ` cfv 5357  opp_rcoppr 14310  LIdealclidl 14741  2Idealc2idl 14773

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-dm 4764  df-iota 5317  df-fv 5365  df-2idl 14774

This theorem is referenced by:  2idlval  14776  2idlelb  14779  2idllidld  14780  2idlridld  14781  2idlbas  14789