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Theorem o1mptrcl 13527
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 13434 . . . . 5  |-  ( ( x  e.  A  |->  B )  e.  O(1)  ->  (
x  e.  A  |->  B ) : dom  (
x  e.  A  |->  B ) --> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> CC )
4 o1add2.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
54ralrimiva 2868 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5487 . . . . . 6  |-  ( A. x  e.  A  B  e.  V  ->  dom  (
x  e.  A  |->  B )  =  A )
75, 6syl 16 . . . . 5  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
87feq2d 5700 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> CC  <->  ( x  e.  A  |->  B ) : A --> CC ) )
93, 8mpbid 210 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> CC )
10 eqid 2454 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 6028 . . 3  |-  ( A. x  e.  A  B  e.  CC  <->  ( x  e.  A  |->  B ) : A --> CC )
129, 11sylibr 212 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
1312r19.21bi 2823 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    |-> cmpt 4497   dom cdm 4988   -->wf 5566   CCcc 9479   O(1)co1 13391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-o1 13395
This theorem is referenced by:  o1le  13557  fsumo1  13708  o1fsum  13709  o1cxp  23502  mulogsum  23915
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