Theorem suplocexprlem2b 7669

Description: Lemma for suplocexpr 7680. 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 5497	. 2 |- ( 2nd ` B ) = ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
3		fo1st 6134	. . . . . 6 |- 1st : _V -onto-> _V
4		fofun 5419	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
5	3, 4	ax-mp 5	. . . . 5 |- Fun 1st
6		npex 7428	. . . . . 6 |- P. e. _V
7	6	ssex 4124	. . . . 5 |- ( A C_ P. -> A e. _V )
8		funimaexg 5280	. . . . 5 |- ( ( Fun 1st /\ A e. _V ) -> ( 1st " A ) e. _V )
9	5, 7, 8	sylancr 412	. . . 4 |- ( A C_ P. -> ( 1st " A ) e. _V )
10		uniexg 4422	. . . 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 7318	. . . 4 |- Q. e. _V
13	12	rabex 4131	. . 3 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
14		op2ndg 6128	. . 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 411	. 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 2215	1 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   C_ wss 3121   <.cop 3584   U.cuni 3794   |^|cint 3829   class class class wbr 3987   "cima 4612   Fun wfun 5190   -onto->wfo 5194   ` cfv 5196   1stc1st 6115   2ndc2nd 6116   Q.cnq 7235   <Q cltq 7240   P.cnp 7246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6117  df-2nd 6118  df-qs 6517  df-ni 7259  df-nqqs 7303  df-inp 7421

This theorem is referenced by:  suplocexprlemmu  7673  suplocexprlemru  7674  suplocexprlemdisj  7675  suplocexprlemloc  7676  suplocexprlemex  7677  suplocexprlemub  7678