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Theorem lo1mptrcl 13201
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 13098 . . . . 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 2822 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5433 . . . . . 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 5645 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR  <->  ( x  e.  A  |->  B ) : A --> RR ) )
93, 8mpbid 210 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
10 eqid 2451 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 5963 . . 3  |-  ( A. x  e.  A  B  e.  RR  <->  ( x  e.  A  |->  B ) : A --> RR )
129, 11sylibr 212 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  RR )
1312r19.21bi 2910 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    |-> cmpt 4448   dom cdm 4938   -->wf 5512   RRcr 9382   <_O(1)clo1 13067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-pm 7317  df-lo1 13071
This theorem is referenced by:  lo1add  13206  lo1mul  13207  lo1mul2  13208  lo1sub  13210  lo1le  13231
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