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Theorem suplocexprlem2b 7529
Description: Lemma for suplocexpr 7540. 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
StepHypRef Expression
1 suplocexprlem2b.b . . 3  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
21fveq2i 5424 . 2  |-  ( 2nd `  B )  =  ( 2nd `  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >. )
3 fo1st 6055 . . . . . 6  |-  1st : _V -onto-> _V
4 fofun 5346 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
53, 4ax-mp 5 . . . . 5  |-  Fun  1st
6 npex 7288 . . . . . 6  |-  P.  e.  _V
76ssex 4065 . . . . 5  |-  ( A 
C_  P.  ->  A  e. 
_V )
8 funimaexg 5207 . . . . 5  |-  ( ( Fun  1st  /\  A  e. 
_V )  ->  ( 1st " A )  e. 
_V )
95, 7, 8sylancr 410 . . . 4  |-  ( A 
C_  P.  ->  ( 1st " A )  e.  _V )
10 uniexg 4361 . . . 4  |-  ( ( 1st " A )  e.  _V  ->  U. ( 1st " A )  e. 
_V )
119, 10syl 14 . . 3  |-  ( A 
C_  P.  ->  U. ( 1st " A )  e. 
_V )
12 nqex 7178 . . . 4  |-  Q.  e.  _V
1312rabex 4072 . . 3  |-  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u }  e.  _V
14 op2ndg 6049 . . 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 } )
1511, 13, 14sylancl 409 . 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 } )
162, 15syl5eq 2184 1  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   E.wrex 2417   {crab 2420   _Vcvv 2686    C_ wss 3071   <.cop 3530   U.cuni 3736   |^|cint 3771   class class class wbr 3929   "cima 4542   Fun wfun 5117   -onto->wfo 5121   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095    <Q cltq 7100   P.cnp 7106
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-inp 7281
This theorem is referenced by:  suplocexprlemmu  7533  suplocexprlemru  7534  suplocexprlemdisj  7535  suplocexprlemloc  7536  suplocexprlemex  7537  suplocexprlemub  7538
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