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Theorem lo1mptrcl 13672
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 13569 . . . . 5  |-  ( ( x  e.  A  |->  B )  e.  <_O(1)  -> 
( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR )
31, 2syl 17 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR )
4 o1add2.1 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
54ralrimiva 2839 . . . . . 6  |-  ( ph  ->  A. x  e.  A  B  e.  V )
6 dmmptg 5347 . . . . . 6  |-  ( A. x  e.  A  B  e.  V  ->  dom  (
x  e.  A  |->  B )  =  A )
75, 6syl 17 . . . . 5  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
87feq2d 5729 . . . 4  |-  ( ph  ->  ( ( x  e.  A  |->  B ) : dom  ( x  e.  A  |->  B ) --> RR  <->  ( x  e.  A  |->  B ) : A --> RR ) )
93, 8mpbid 213 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> RR )
10 eqid 2422 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
1110fmpt 6054 . . 3  |-  ( A. x  e.  A  B  e.  RR  <->  ( x  e.  A  |->  B ) : A --> RR )
129, 11sylibr 215 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  RR )
1312r19.21bi 2794 1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775    |-> cmpt 4479   dom cdm 4849   -->wf 5593   RRcr 9538   <_O(1)clo1 13538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-pm 7479  df-lo1 13542
This theorem is referenced by:  lo1add  13677  lo1mul  13678  lo1mul2  13679  lo1sub  13681  lo1le  13702
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