Theorem suplocexprlem2b 7712

Description: Lemma for suplocexpr 7723. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

Hypothesis

Ref	Expression
suplocexprlem2b.b	|- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.

Assertion

Ref	Expression
suplocexprlem2b	|- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )

Proof of Theorem suplocexprlem2b

Step	Hyp	Ref	Expression
1		suplocexprlem2b.b	. . 3 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
2	1	fveq2i 5518	. 2 |- ( 2nd ` B ) = ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
3		fo1st 6157	. . . . . 6 |- 1st : _V -onto-> _V
4		fofun 5439	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
5	3, 4	ax-mp 5	. . . . 5 |- Fun 1st
6		npex 7471	. . . . . 6 |- P. e. _V
7	6	ssex 4140	. . . . 5 |- ( A C_ P. -> A e. _V )
8		funimaexg 5300	. . . . 5 |- ( ( Fun 1st /\ A e. _V ) -> ( 1st " A ) e. _V )
9	5, 7, 8	sylancr 414	. . . 4 |- ( A C_ P. -> ( 1st " A ) e. _V )
10		uniexg 4439	. . . 4 |- ( ( 1st " A ) e. _V -> U. ( 1st " A ) e. _V )
11	9, 10	syl 14	. . 3 |- ( A C_ P. -> U. ( 1st " A ) e. _V )
12		nqex 7361	. . . 4 |- Q. e. _V
13	12	rabex 4147	. . 3 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
14		op2ndg 6151	. . 3 |- ( ( U. ( 1st " A ) e. _V /\ { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V ) -> ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
15	11, 13, 14	sylancl 413	. 2 |- ( A C_ P. -> ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
16	2, 15	eqtrid 2222	1 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   _Vcvv 2737   C_ wss 3129   <.cop 3595   U.cuni 3809   |^|cint 3844   class class class wbr 4003   "cima 4629   Fun wfun 5210   -onto->wfo 5214   ` cfv 5216   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   <Q cltq 7283   P.cnp 7289

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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-qs 6540  df-ni 7302  df-nqqs 7346  df-inp 7464

This theorem is referenced by:  suplocexprlemmu  7716  suplocexprlemru  7717  suplocexprlemdisj  7718  suplocexprlemloc  7719  suplocexprlemex  7720  suplocexprlemub  7721