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Theorem lo1mptrcl 12344
Description: Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
o1add2.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
lo1mptrcl.3  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )
Assertion
Ref Expression
lo1mptrcl  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem lo1mptrcl
StepHypRef Expression
1 lo1mptrcl.3 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )
2 lo1f 12241 . . . . 5  |-  ( ( x  e.  A  |->  B )  e.  <_ O
( 1 )  -> 
( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR )
31, 2syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR )
4 o1add2.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
54ralrimiva 2734 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5309 . . . . . 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 5523 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR  <->  ( x  e.  A  |->  B ) : A --> RR ) )
93, 8mpbid 202 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
10 eqid 2389 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 5831 . . 3  |-  ( A. x  e.  A  B  e.  RR  <->  ( x  e.  A  |->  B ) : A --> RR )
129, 11sylibr 204 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  RR )
1312r19.21bi 2749 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651    e. cmpt 4209   dom cdm 4820   -->wf 5392   RRcr 8924   <_ O ( 1 )clo1 12210
This theorem is referenced by:  lo1add  12349  lo1mul  12350  lo1mul2  12351  lo1sub  12353  lo1le  12374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-pm 6959  df-lo1 12214
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