Theorem suplocexprlem2b 7827

Description: Lemma for suplocexpr 7838. 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 5579	. 2 |- ( 2nd ` B ) = ( 2nd ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
3		fo1st 6243	. . . . . 6 |- 1st : _V -onto-> _V
4		fofun 5499	. . . . . 6 |- ( 1st : _V -onto-> _V -> Fun 1st )
5	3, 4	ax-mp 5	. . . . 5 |- Fun 1st
6		npex 7586	. . . . . 6 |- P. e. _V
7	6	ssex 4181	. . . . 5 |- ( A C_ P. -> A e. _V )
8		funimaexg 5358	. . . . 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 4486	. . . 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 7476	. . . 4 |- Q. e. _V
13	12	rabex 4188	. . 3 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
14		op2ndg 6237	. . 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 2250	1 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   E.wrex 2485   {crab 2488   _Vcvv 2772   C_ wss 3166   <.cop 3636   U.cuni 3850   |^|cint 3885   class class class wbr 4044   "cima 4678   Fun wfun 5265   -onto->wfo 5269   ` cfv 5271   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   <Q cltq 7398   P.cnp 7404

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-qs 6626  df-ni 7417  df-nqqs 7461  df-inp 7579

This theorem is referenced by:  suplocexprlemmu  7831  suplocexprlemru  7832  suplocexprlemdisj  7833  suplocexprlemloc  7834  suplocexprlemex  7835  suplocexprlemub  7836