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Theorem 0pledm 22508
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  ->  ( 0p  oR  <_  F  <->  ( A  X.  { 0 } )  oR  <_  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 3687 . . . 4  |-  ( A 
C_  CC  <->  ( CC  i^i  A )  =  A )
31, 2sylib 199 . . 3  |-  ( ph  ->  ( CC  i^i  A
)  =  A )
43raleqdv 3038 . 2  |-  ( ph  ->  ( A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
5 0cn 9634 . . . . . 6  |-  0  e.  CC
6 fnconstg 5788 . . . . . 6  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
75, 6ax-mp 5 . . . . 5  |-  ( CC 
X.  { 0 } )  Fn  CC
8 df-0p 22505 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
98fneq1i 5688 . . . . 5  |-  ( 0p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
107, 9mpbir 212 . . . 4  |-  0p  Fn  CC
1110a1i 11 . . 3  |-  ( ph  ->  0p  Fn  CC )
12 0pledm.2 . . 3  |-  ( ph  ->  F  Fn  A )
13 cnex 9619 . . . 4  |-  CC  e.  _V
1413a1i 11 . . 3  |-  ( ph  ->  CC  e.  _V )
15 ssexg 4571 . . . 4  |-  ( ( A  C_  CC  /\  CC  e.  _V )  ->  A  e.  _V )
161, 13, 15sylancl 666 . . 3  |-  ( ph  ->  A  e.  _V )
17 eqid 2429 . . 3  |-  ( CC 
i^i  A )  =  ( CC  i^i  A
)
18 0pval 22506 . . . 4  |-  ( x  e.  CC  ->  (
0p `  x
)  =  0 )
1918adantl 467 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0p `  x )  =  0 )
20 eqidd 2430 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
2111, 12, 14, 16, 17, 19, 20ofrfval 6553 . 2  |-  ( ph  ->  ( 0p  oR  <_  F  <->  A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )
) )
22 fnconstg 5788 . . . . 5  |-  ( 0  e.  CC  ->  ( A  X.  { 0 } )  Fn  A )
235, 22ax-mp 5 . . . 4  |-  ( A  X.  { 0 } )  Fn  A
2423a1i 11 . . 3  |-  ( ph  ->  ( A  X.  {
0 } )  Fn  A )
25 inidm 3677 . . 3  |-  ( A  i^i  A )  =  A
26 c0ex 9636 . . . . 5  |-  0  e.  _V
2726fvconst2 6135 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2827adantl 467 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2924, 12, 16, 16, 25, 28, 20ofrfval 6553 . 2  |-  ( ph  ->  ( ( A  X.  { 0 } )  oR  <_  F  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
304, 21, 293bitr4d 288 1  |-  ( ph  ->  ( 0p  oR  <_  F  <->  ( A  X.  { 0 } )  oR  <_  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    i^i cin 3441    C_ wss 3442   {csn 4002   class class class wbr 4426    X. cxp 4852    Fn wfn 5596   ` cfv 5601    oRcofr 6544   CCcc 9536   0cc0 9538    <_ cle 9675   0pc0p 22504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-cnex 9594  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-mulcl 9600  ax-i2m1 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ofr 6546  df-0p 22505
This theorem is referenced by:  xrge0f  22566  itg20  22572  itg2const  22575  i1fibl  22642  itgitg1  22643  ftc1anclem5  31725  ftc1anclem7  31727
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