Theorem recexprlemell 7979

Description: Membership in the lower cut of B. Lemma for recexpr 7995. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemell	|- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem recexprlemell

Step	Hyp	Ref	Expression
1		elex 2833	. 2 |- ( C e. ( 1st ` B ) -> C e. _V )
2		ltrelnq 7722	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4822	. . . . . 6 |- ( C <Q y -> ( C e. Q. /\ y e. Q. ) )
4	3	simpld 112	. . . . 5 |- ( C <Q y -> C e. Q. )
5		elex 2833	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( C <Q y -> C e. _V )
7	6	adantr 276	. . 3 |- ( ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
8	7	exlimiv 1651	. 2 |- ( E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
9		breq1 4128	. . . . 5 |- ( x = C -> ( x <Q y <-> C <Q y ) )
10	9	anbi1d 469	. . . 4 |- ( x = C -> ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
11	10	exbidv 1878	. . 3 |- ( x = C -> ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
12		recexpr.1	. . . . 5 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
13	12	fveq2i 5693	. . . 4 |- ( 1st ` B ) = ( 1st ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. )
14		nqex 7720	. . . . . 6 |- Q. e. _V
15	2	brel 4822	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
16	15	simpld 112	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
17	16	adantr 276	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
18	17	exlimiv 1651	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3323	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4266	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4822	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
22	21	simprd 114	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
23	22	adantr 276	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
24	23	exlimiv 1651	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3323	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4266	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op1st 6370	. . . 4 |- ( 1st ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
28	13, 27	eqtri 2259	. . 3 |- ( 1st ` B ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
29	11, 28	elab2g 2973	. 2 |- ( C e. _V -> ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
30	1, 8, 29	pm5.21nii 716	1 |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   _Vcvv 2821   <.cop 3708   class class class wbr 4125   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   *Qcrq 7641   <Q cltq 7642

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-in1 623  ax-in2 624  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-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  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-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-qs 6803  df-ni 7661  df-nqqs 7705  df-ltnqqs 7710

This theorem is referenced by:  recexprlemm  7981  recexprlemopl  7982  recexprlemlol  7983  recexprlemdisj  7987  recexprlemloc  7988  recexprlem1ssl  7990  recexprlemss1l  7992