Theorem suplocexprlem2b 7776

Description: Lemma for suplocexpr 7787. 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 5558	. 2 |- ( 2nd ` B ) = ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
3		fo1st 6212	. . . . . 6 |- 1st : _V -onto-> _V
4		fofun 5478	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
5	3, 4	ax-mp 5	. . . . 5 |- Fun 1st
6		npex 7535	. . . . . 6 |- P. e. _V
7	6	ssex 4167	. . . . 5 |- ( A C_ P. -> A e. _V )
8		funimaexg 5339	. . . . 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 4471	. . . 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 7425	. . . 4 |- Q. e. _V
13	12	rabex 4174	. . 3 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
14		op2ndg 6206	. . 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 2238	1 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   E.wrex 2473   {crab 2476   _Vcvv 2760   C_ wss 3154   <.cop 3622   U.cuni 3836   |^|cint 3871   class class class wbr 4030   "cima 4663   Fun wfun 5249   -onto->wfo 5253   ` cfv 5255   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   <Q cltq 7347   P.cnp 7353

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-qs 6595  df-ni 7366  df-nqqs 7410  df-inp 7528

This theorem is referenced by:  suplocexprlemmu  7780  suplocexprlemru  7781  suplocexprlemdisj  7782  suplocexprlemloc  7783  suplocexprlemex  7784  suplocexprlemub  7785