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Theorem lo1mptrcl 12095
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 11992 . . . . 5  |-  ( ( x  e.  A  |->  B )  e.  <_ O
( 1 )  -> 
( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR )
31, 2syl 15 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR )
4 o1add2.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
54ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5170 . . . . . 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 5380 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR  <->  ( x  e.  A  |->  B ) : A --> RR ) )
93, 8mpbid 201 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
10 eqid 2283 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 5681 . . 3  |-  ( A. x  e.  A  B  e.  RR  <->  ( x  e.  A  |->  B ) : A --> RR )
129, 11sylibr 203 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  RR )
1312r19.21bi 2641 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   dom cdm 4689   -->wf 5251   RRcr 8736   <_ O ( 1 )clo1 11961
This theorem is referenced by:  lo1add  12100  lo1mul  12101  lo1mul2  12102  lo1sub  12104  lo1le  12125
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-pm 6775  df-lo1 11965
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