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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
StepHypRef Expression
1 suplocexprlem2b.b . . 3  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
21fveq2i 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 )
53, 4ax-mp 5 . . . . 5  |-  Fun  1st
6 npex 7428 . . . . . 6  |-  P.  e.  _V
76ssex 4124 . . . . 5  |-  ( A 
C_  P.  ->  A  e. 
_V )
8 funimaexg 5280 . . . . 5  |-  ( ( Fun  1st  /\  A  e. 
_V )  ->  ( 1st " A )  e. 
_V )
95, 7, 8sylancr 412 . . . 4  |-  ( A 
C_  P.  ->  ( 1st " A )  e.  _V )
10 uniexg 4422 . . . 4  |-  ( ( 1st " A )  e.  _V  ->  U. ( 1st " A )  e. 
_V )
119, 10syl 14 . . 3  |-  ( A 
C_  P.  ->  U. ( 1st " A )  e. 
_V )
12 nqex 7318 . . . 4  |-  Q.  e.  _V
1312rabex 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 } )
1511, 13, 14sylancl 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 } )
162, 15eqtrid 2215 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 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
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