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Theorem 0pledm 19028
Description: Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
0pledm.1  |-  ( ph  ->  A  C_  CC )
0pledm.2  |-  ( ph  ->  F  Fn  A )
Assertion
Ref Expression
0pledm  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  ( A  X.  { 0 } )  o R  <_  F
) )

Proof of Theorem 0pledm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0pledm.1 . . . 4  |-  ( ph  ->  A  C_  CC )
2 sseqin2 3388 . . . 4  |-  ( A 
C_  CC  <->  ( CC  i^i  A )  =  A )
31, 2sylib 188 . . 3  |-  ( ph  ->  ( CC  i^i  A
)  =  A )
43raleqdv 2742 . 2  |-  ( ph  ->  ( A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
5 0cn 8831 . . . . . 6  |-  0  e.  CC
6 fnconstg 5429 . . . . . 6  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
75, 6ax-mp 8 . . . . 5  |-  ( CC 
X.  { 0 } )  Fn  CC
8 df-0p 19025 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
98fneq1i 5338 . . . . 5  |-  ( 0 p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
107, 9mpbir 200 . . . 4  |-  0 p  Fn  CC
1110a1i 10 . . 3  |-  ( ph  ->  0 p  Fn  CC )
12 0pledm.2 . . 3  |-  ( ph  ->  F  Fn  A )
13 cnex 8818 . . . 4  |-  CC  e.  _V
1413a1i 10 . . 3  |-  ( ph  ->  CC  e.  _V )
15 ssexg 4160 . . . 4  |-  ( ( A  C_  CC  /\  CC  e.  _V )  ->  A  e.  _V )
161, 13, 15sylancl 643 . . 3  |-  ( ph  ->  A  e.  _V )
17 eqid 2283 . . 3  |-  ( CC 
i^i  A )  =  ( CC  i^i  A
)
18 0pval 19026 . . . 4  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
1918adantl 452 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
20 eqidd 2284 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
2111, 12, 14, 16, 17, 19, 20ofrfval 6086 . 2  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )
) )
22 fnconstg 5429 . . . . 5  |-  ( 0  e.  CC  ->  ( A  X.  { 0 } )  Fn  A )
235, 22ax-mp 8 . . . 4  |-  ( A  X.  { 0 } )  Fn  A
2423a1i 10 . . 3  |-  ( ph  ->  ( A  X.  {
0 } )  Fn  A )
25 inidm 3378 . . 3  |-  ( A  i^i  A )  =  A
26 c0ex 8832 . . . . 5  |-  0  e.  _V
2726fvconst2 5729 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2827adantl 452 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2924, 12, 16, 16, 25, 28, 20ofrfval 6086 . 2  |-  ( ph  ->  ( ( A  X.  { 0 } )  o R  <_  F  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
304, 21, 293bitr4d 276 1  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  ( A  X.  { 0 } )  o R  <_  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023    X. cxp 4687    Fn wfn 5250   ` cfv 5255    o Rcofr 6077   CCcc 8735   0cc0 8737    <_ cle 8868   0 pc0p 19024
This theorem is referenced by:  xrge0f  19086  itg20  19092  itg2const  19095  i1fibl  19162  itgitg1  19163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-cnex 8793  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ofr 6079  df-0p 19025
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