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Theorem lo1mptrcl 12405
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 12302 . . . . 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 2781 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5359 . . . . . 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 5573 . . . 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 2435 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 5882 . . 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 2796 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258   dom cdm 4870   -->wf 5442   RRcr 8979   <_ O ( 1 )clo1 12271
This theorem is referenced by:  lo1add  12410  lo1mul  12411  lo1mul2  12412  lo1sub  12414  lo1le  12435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-pm 7013  df-lo1 12275
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