Theorem 2idlmex 14649

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 4883	. . . 4 |- Rel ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
2		df-2idl 14648	. . . . 5 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3	2	releqi 4833	. . . 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 2299	. . . 4 |- ( U e. T <-> U e. (2Ideal ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. T -> U e. (2Ideal ` W ) )
8		relelfvdm 5702	. . 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 2827	1 |- ( U e. T -> W e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   |-> cmpt 4171   dom cdm 4749   Rel wrel 4754   ` cfv 5352  opp_rcoppr 14211  LIdealclidl 14615  2Idealc2idl 14647

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-xp 4755  df-rel 4756  df-dm 4759  df-iota 5312  df-fv 5360  df-2idl 14648

This theorem is referenced by:  2idlval  14650  2idlelb  14653  2idllidld  14654  2idlridld  14655  2idlbas  14663