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Theorem 0pledm 19044
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 3401 . . . 4  |-  ( A 
C_  CC  <->  ( CC  i^i  A )  =  A )
31, 2sylib 188 . . 3  |-  ( ph  ->  ( CC  i^i  A
)  =  A )
43raleqdv 2755 . 2  |-  ( ph  ->  ( A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
5 0cn 8847 . . . . . 6  |-  0  e.  CC
6 fnconstg 5445 . . . . . 6  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
75, 6ax-mp 8 . . . . 5  |-  ( CC 
X.  { 0 } )  Fn  CC
8 df-0p 19041 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
98fneq1i 5354 . . . . 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 8834 . . . 4  |-  CC  e.  _V
1413a1i 10 . . 3  |-  ( ph  ->  CC  e.  _V )
15 ssexg 4176 . . . 4  |-  ( ( A  C_  CC  /\  CC  e.  _V )  ->  A  e.  _V )
161, 13, 15sylancl 643 . . 3  |-  ( ph  ->  A  e.  _V )
17 eqid 2296 . . 3  |-  ( CC 
i^i  A )  =  ( CC  i^i  A
)
18 0pval 19042 . . . 4  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
1918adantl 452 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
20 eqidd 2297 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
2111, 12, 14, 16, 17, 19, 20ofrfval 6102 . 2  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )
) )
22 fnconstg 5445 . . . . 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 3391 . . 3  |-  ( A  i^i  A )  =  A
26 c0ex 8848 . . . . 5  |-  0  e.  _V
2726fvconst2 5745 . . . 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 6102 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   class class class wbr 4039    X. cxp 4703    Fn wfn 5266   ` cfv 5271    o Rcofr 6093   CCcc 8751   0cc0 8753    <_ cle 8884   0 pc0p 19040
This theorem is referenced by:  xrge0f  19102  itg20  19108  itg2const  19111  i1fibl  19178  itgitg1  19179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-cnex 8809  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ofr 6095  df-0p 19041
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