MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  o1mptrcl Unicode version

Theorem o1mptrcl 12092
Description: Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
o1add2.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
o1mptrcl.3  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O
( 1 ) )
Assertion
Ref Expression
o1mptrcl  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem o1mptrcl
StepHypRef Expression
1 o1mptrcl.3 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O
( 1 ) )
2 o1f 11999 . . . . 5  |-  ( ( x  e.  A  |->  B )  e.  O ( 1 )  ->  (
x  e.  A  |->  B ) :  dom  (  x  e.  A  |->  B ) --> CC )
31, 2syl 15 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B ) :  dom  (  x  e.  A  |->  B ) --> CC )
4 o1add2.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
54ralrimiva 2627 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5168 . . . . . 6  |-  ( A. x  e.  A  B  e.  V  ->  dom  (  x  e.  A  |->  B )  =  A )
75, 6syl 15 . . . . 5  |-  ( ph  ->  dom  (  x  e.  A  |->  B )  =  A )
87feq2d 5346 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) :  dom  (  x  e.  A  |->  B ) --> CC  <->  ( x  e.  A  |->  B ) : A --> CC ) )
93, 8mpbid 201 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> CC )
10 eqid 2284 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 5643 . . 3  |-  ( A. x  e.  A  B  e.  CC  <->  ( x  e.  A  |->  B ) : A --> CC )
129, 11sylibr 203 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
1312r19.21bi 2642 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544    e. cmpt 4078    dom cdm 4688   -->wf 5217   CCcc 8731   O ( 1 )co1 11956
This theorem is referenced by:  o1le  12122  fsumo1  12266  o1fsum  12267  o1cxp  20265  mulogsum  20677
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-pm 6771  df-o1 11960
  Copyright terms: Public domain W3C validator