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Theorem 0pledm 21287
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 3680 . . . 4  |-  ( A 
C_  CC  <->  ( CC  i^i  A )  =  A )
31, 2sylib 196 . . 3  |-  ( ph  ->  ( CC  i^i  A
)  =  A )
43raleqdv 3029 . 2  |-  ( ph  ->  ( A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
5 0cn 9492 . . . . . 6  |-  0  e.  CC
6 fnconstg 5709 . . . . . 6  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
75, 6ax-mp 5 . . . . 5  |-  ( CC 
X.  { 0 } )  Fn  CC
8 df-0p 21284 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
98fneq1i 5616 . . . . 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 9477 . . . 4  |-  CC  e.  _V
1413a1i 11 . . 3  |-  ( ph  ->  CC  e.  _V )
15 ssexg 4549 . . . 4  |-  ( ( A  C_  CC  /\  CC  e.  _V )  ->  A  e.  _V )
161, 13, 15sylancl 662 . . 3  |-  ( ph  ->  A  e.  _V )
17 eqid 2454 . . 3  |-  ( CC 
i^i  A )  =  ( CC  i^i  A
)
18 0pval 21285 . . . 4  |-  ( x  e.  CC  ->  (
0p `  x
)  =  0 )
1918adantl 466 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0p `  x )  =  0 )
20 eqidd 2455 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
2111, 12, 14, 16, 17, 19, 20ofrfval 6441 . 2  |-  ( ph  ->  ( 0p  oR  <_  F  <->  A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )
) )
22 fnconstg 5709 . . . . 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 3670 . . 3  |-  ( A  i^i  A )  =  A
26 c0ex 9494 . . . . 5  |-  0  e.  _V
2726fvconst2 6045 . . . 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 6441 . 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 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    i^i cin 3438    C_ wss 3439   {csn 3988   class class class wbr 4403    X. cxp 4949    Fn wfn 5524   ` cfv 5529    oRcofr 6432   CCcc 9394   0cc0 9396    <_ cle 9533   0pc0p 21283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-cnex 9452  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-mulcl 9458  ax-i2m1 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ofr 6434  df-0p 21284
This theorem is referenced by:  xrge0f  21345  itg20  21351  itg2const  21354  i1fibl  21421  itgitg1  21422  ftc1anclem5  28639  ftc1anclem7  28641
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