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Theorem 0pledm 21907
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 3717 . . . 4  |-  ( A 
C_  CC  <->  ( CC  i^i  A )  =  A )
31, 2sylib 196 . . 3  |-  ( ph  ->  ( CC  i^i  A
)  =  A )
43raleqdv 3064 . 2  |-  ( ph  ->  ( A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
5 0cn 9589 . . . . . 6  |-  0  e.  CC
6 fnconstg 5773 . . . . . 6  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
75, 6ax-mp 5 . . . . 5  |-  ( CC 
X.  { 0 } )  Fn  CC
8 df-0p 21904 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
98fneq1i 5675 . . . . 5  |-  ( 0p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
107, 9mpbir 209 . . . 4  |-  0p  Fn  CC
1110a1i 11 . . 3  |-  ( ph  ->  0p  Fn  CC )
12 0pledm.2 . . 3  |-  ( ph  ->  F  Fn  A )
13 cnex 9574 . . . 4  |-  CC  e.  _V
1413a1i 11 . . 3  |-  ( ph  ->  CC  e.  _V )
15 ssexg 4593 . . . 4  |-  ( ( A  C_  CC  /\  CC  e.  _V )  ->  A  e.  _V )
161, 13, 15sylancl 662 . . 3  |-  ( ph  ->  A  e.  _V )
17 eqid 2467 . . 3  |-  ( CC 
i^i  A )  =  ( CC  i^i  A
)
18 0pval 21905 . . . 4  |-  ( x  e.  CC  ->  (
0p `  x
)  =  0 )
1918adantl 466 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0p `  x )  =  0 )
20 eqidd 2468 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
2111, 12, 14, 16, 17, 19, 20ofrfval 6533 . 2  |-  ( ph  ->  ( 0p  oR  <_  F  <->  A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )
) )
22 fnconstg 5773 . . . . 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 3707 . . 3  |-  ( A  i^i  A )  =  A
26 c0ex 9591 . . . . 5  |-  0  e.  _V
2726fvconst2 6117 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2827adantl 466 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2924, 12, 16, 16, 25, 28, 20ofrfval 6533 . 2  |-  ( ph  ->  ( ( A  X.  { 0 } )  oR  <_  F  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
304, 21, 293bitr4d 285 1  |-  ( ph  ->  ( 0p  oR  <_  F  <->  ( A  X.  { 0 } )  oR  <_  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   {csn 4027   class class class wbr 4447    X. cxp 4997    Fn wfn 5583   ` cfv 5588    oRcofr 6524   CCcc 9491   0cc0 9493    <_ cle 9630   0pc0p 21903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-cnex 9549  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-mulcl 9555  ax-i2m1 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ofr 6526  df-0p 21904
This theorem is referenced by:  xrge0f  21965  itg20  21971  itg2const  21974  i1fibl  22041  itgitg1  22042  ftc1anclem5  29947  ftc1anclem7  29949
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