Theorem suplocexprlem2b 7977

Description: Lemma for suplocexpr 7988. 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 5651	. 2 |- ( 2nd ` B ) = ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
3		fo1st 6329	. . . . . 6 |- 1st : _V -onto-> _V
4		fofun 5569	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
5	3, 4	ax-mp 5	. . . . 5 |- Fun 1st
6		npex 7736	. . . . . 6 |- P. e. _V
7	6	ssex 4231	. . . . 5 |- ( A C_ P. -> A e. _V )
8		funimaexg 5421	. . . . 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 4542	. . . 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 7626	. . . 4 |- Q. e. _V
13	12	rabex 4239	. . 3 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
14		op2ndg 6323	. . 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 2276	1 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   E.wrex 2512   {crab 2515   _Vcvv 2803   C_ wss 3201   <.cop 3676   U.cuni 3898   |^|cint 3933   class class class wbr 4093   "cima 4734   Fun wfun 5327   -onto->wfo 5331   ` cfv 5333   1stc1st 6310   2ndc2nd 6311   Q.cnq 7543   <Q cltq 7548   P.cnp 7554

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-1st 6312  df-2nd 6313  df-qs 6751  df-ni 7567  df-nqqs 7611  df-inp 7729

This theorem is referenced by:  suplocexprlemmu  7981  suplocexprlemru  7982  suplocexprlemdisj  7983  suplocexprlemloc  7984  suplocexprlemex  7985  suplocexprlemub  7986