Theorem ltexprlemell 7861

Description: Element in lower cut of the constructed difference. Lemma for ltexpri 7876. (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
ltexprlemell	|- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )

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

Proof of Theorem ltexprlemell

Step	Hyp	Ref	Expression
1		oveq2 6036	. . . . 5 |- ( x = q -> ( y +Q x ) = ( y +Q q ) )
2	1	eleq1d 2300	. . . 4 |- ( x = q -> ( ( y +Q x ) e. ( 1st ` B ) <-> ( y +Q q ) e. ( 1st ` B ) ) )
3	2	anbi2d 464	. . 3 |- ( x = q -> ( ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) <-> ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
4	3	exbidv 1873	. 2 |- ( x = q -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` 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 5651	. . 3 |- ( 1st ` C ) = ( 1st ` <. { 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 7626	. . . . 5 |- Q. e. _V
8	7	rabex 4239	. . . 4 |- { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } e. _V
9	7	rabex 4239	. . . 4 |- { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } e. _V
10	8, 9	op1st 6318	. . 3 |- ( 1st ` <. { 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. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) }
11	6, 10	eqtri 2252	. 2 |- ( 1st ` C ) = { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) }
12	4, 11	elrab2 2966	1 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   {crab 2515   <.cop 3676   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   Q.cnq 7543   +Q cplq 7545

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 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-1st 6312  df-qs 6751  df-ni 7567  df-nqqs 7611

This theorem is referenced by:  ltexprlemm  7863  ltexprlemopl  7864  ltexprlemlol  7865  ltexprlemdisj  7869  ltexprlemloc  7870  ltexprlemfl  7872  ltexprlemrl  7873