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Theorem suplocexprlem2b 7715
Description: Lemma for suplocexpr 7726. 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 5520 . 2  |-  ( 2nd `  B )  =  ( 2nd `  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >. )
3 fo1st 6160 . . . . . 6  |-  1st : _V -onto-> _V
4 fofun 5441 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
53, 4ax-mp 5 . . . . 5  |-  Fun  1st
6 npex 7474 . . . . . 6  |-  P.  e.  _V
76ssex 4142 . . . . 5  |-  ( A 
C_  P.  ->  A  e. 
_V )
8 funimaexg 5302 . . . . 5  |-  ( ( Fun  1st  /\  A  e. 
_V )  ->  ( 1st " A )  e. 
_V )
95, 7, 8sylancr 414 . . . 4  |-  ( A 
C_  P.  ->  ( 1st " A )  e.  _V )
10 uniexg 4441 . . . 4  |-  ( ( 1st " A )  e.  _V  ->  U. ( 1st " A )  e. 
_V )
119, 10syl 14 . . 3  |-  ( A 
C_  P.  ->  U. ( 1st " A )  e. 
_V )
12 nqex 7364 . . . 4  |-  Q.  e.  _V
1312rabex 4149 . . 3  |-  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u }  e.  _V
14 op2ndg 6154 . . 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 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 } )
162, 15eqtrid 2222 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 1353    e. wcel 2148   E.wrex 2456   {crab 2459   _Vcvv 2739    C_ wss 3131   <.cop 3597   U.cuni 3811   |^|cint 3846   class class class wbr 4005   "cima 4631   Fun wfun 5212   -onto->wfo 5216   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281    <Q cltq 7286   P.cnp 7292
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349  df-inp 7467
This theorem is referenced by:  suplocexprlemmu  7719  suplocexprlemru  7720  suplocexprlemdisj  7721  suplocexprlemloc  7722  suplocexprlemex  7723  suplocexprlemub  7724
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