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Theorem o1mptrcl 12098
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 12005 . . . . 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 2628 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5172 . . . . . 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 5382 . . . 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 2285 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 5683 . . 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 2643 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545    e. cmpt 4079   dom cdm 4691   -->wf 5253   CCcc 8737   O ( 1 )co1 11962
This theorem is referenced by:  o1le  12128  fsumo1  12272  o1fsum  12273  o1cxp  20271  mulogsum  20683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-pm 6777  df-o1 11966
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