Theorem ltexprlemelu 7600

Description: Element in upper cut of the constructed difference. Lemma for ltexpri 7614. (Contributed by Jim Kingdon, 21-Dec-2019.)

Hypothesis

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

Assertion

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

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

Proof of Theorem ltexprlemelu

Step	Hyp	Ref	Expression
1		oveq2 5885	. . . . 5 |- ( x = r -> ( y +Q x ) = ( y +Q r ) )
2	1	eleq1d 2246	. . . 4 |- ( x = r -> ( ( y +Q x ) e. ( 2nd ` B ) <-> ( y +Q r ) e. ( 2nd ` B ) ) )
3	2	anbi2d 464	. . 3 |- ( x = r -> ( ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) <-> ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
4	3	exbidv 1825	. 2 |- ( x = r -> ( E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) <-> E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
5		ltexprlem.1	. . . 4 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
6	5	fveq2i 5520	. . 3 |- ( 2nd ` C ) = ( 2nd ` <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >. )
7		nqex 7364	. . . . 5 |- Q. e. _V
8	7	rabex 4149	. . . 4 |- { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } e. _V
9	7	rabex 4149	. . . 4 |- { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } e. _V
10	8, 9	op2nd 6150	. . 3 |- ( 2nd ` <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >. ) = { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) }
11	6, 10	eqtri 2198	. 2 |- ( 2nd ` C ) = { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) }
12	4, 11	elrab2 2898	1 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   {crab 2459   <.cop 3597   ` cfv 5218  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   +Q cplq 7283

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349

This theorem is referenced by:  ltexprlemm  7601  ltexprlemopu  7604  ltexprlemupu  7605  ltexprlemdisj  7607  ltexprlemloc  7608  ltexprlemfu  7612  ltexprlemru  7613