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Theorem trnij 25615
Description: A translation is 1-1-onto. (Contributed by FL, 17-Feb-2008.)
Hypothesis
Ref Expression
trnij.1  |-  F  =  ( x  e.  RR  |->  ( x  +  A
) )
Assertion
Ref Expression
trnij  |-  ( A  e.  RR  ->  F : RR -1-1-onto-> RR )
Distinct variable groups:    x, A    x, F

Proof of Theorem trnij
Dummy variables  u  v  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trnij.1 . . . . . . . 8  |-  F  =  ( x  e.  RR  |->  ( x  +  A
) )
21trdom 25613 . . . . . . 7  |-  ( A  e.  RR  ->  dom  F  =  RR )
31funmpt2 5291 . . . . . . 7  |-  Fun  F
42, 3jctil 523 . . . . . 6  |-  ( A  e.  RR  ->  ( Fun  F  /\  dom  F  =  RR ) )
5 df-fn 5258 . . . . . 6  |-  ( F  Fn  RR  <->  ( Fun  F  /\  dom  F  =  RR ) )
64, 5sylibr 203 . . . . 5  |-  ( A  e.  RR  ->  F  Fn  RR )
71trran 25614 . . . . 5  |-  ( A  e.  RR  ->  ran  F  =  RR )
8 eqimss 3230 . . . . . 6  |-  ( ran 
F  =  RR  ->  ran 
F  C_  RR )
98anim2i 552 . . . . 5  |-  ( ( F  Fn  RR  /\  ran  F  =  RR )  ->  ( F  Fn  RR  /\  ran  F  C_  RR ) )
106, 7, 9syl2anc 642 . . . 4  |-  ( A  e.  RR  ->  ( F  Fn  RR  /\  ran  F 
C_  RR ) )
11 df-f 5259 . . . 4  |-  ( F : RR --> RR  <->  ( F  Fn  RR  /\  ran  F  C_  RR ) )
1210, 11sylibr 203 . . 3  |-  ( A  e.  RR  ->  F : RR --> RR )
13 readdcl 8820 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR )  ->  ( x  +  A
)  e.  RR )
1413expcom 424 . . . . . . 7  |-  ( A  e.  RR  ->  (
x  e.  RR  ->  ( x  +  A )  e.  RR ) )
15 readdcl 8820 . . . . . . . 8  |-  ( ( y  e.  RR  /\  A  e.  RR )  ->  ( y  +  A
)  e.  RR )
1615expcom 424 . . . . . . 7  |-  ( A  e.  RR  ->  (
y  e.  RR  ->  ( y  +  A )  e.  RR ) )
1714, 16anim12d 546 . . . . . 6  |-  ( A  e.  RR  ->  (
( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  A )  e.  RR  /\  ( y  +  A )  e.  RR ) ) )
1817imp 418 . . . . 5  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( (
x  +  A )  e.  RR  /\  (
y  +  A )  e.  RR ) )
19 simpr 447 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  ( F `  x )  =  ( F `  y ) )
201fveq1i 5526 . . . . . . . . 9  |-  ( F `
 x )  =  ( ( x  e.  RR  |->  ( x  +  A ) ) `  x )
21 df-mpt 4079 . . . . . . . . . . . 12  |-  ( x  e.  RR  |->  ( x  +  A ) )  =  { <. x ,  u >.  |  (
x  e.  RR  /\  u  =  ( x  +  A ) ) }
22 eleq1 2343 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  (
x  e.  RR  <->  y  e.  RR ) )
2322adantr 451 . . . . . . . . . . . . . 14  |-  ( ( x  =  y  /\  u  =  v )  ->  ( x  e.  RR  <->  y  e.  RR ) )
24 eqeq1 2289 . . . . . . . . . . . . . . 15  |-  ( u  =  v  ->  (
u  =  ( x  +  A )  <->  v  =  ( x  +  A
) ) )
25 oveq1 5865 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
x  +  A )  =  ( y  +  A ) )
2625eqeq2d 2294 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  (
v  =  ( x  +  A )  <->  v  =  ( y  +  A
) ) )
2724, 26sylan9bbr 681 . . . . . . . . . . . . . 14  |-  ( ( x  =  y  /\  u  =  v )  ->  ( u  =  ( x  +  A )  <-> 
v  =  ( y  +  A ) ) )
2823, 27anbi12d 691 . . . . . . . . . . . . 13  |-  ( ( x  =  y  /\  u  =  v )  ->  ( ( x  e.  RR  /\  u  =  ( x  +  A
) )  <->  ( y  e.  RR  /\  v  =  ( y  +  A
) ) ) )
2928cbvopabv 4088 . . . . . . . . . . . 12  |-  { <. x ,  u >.  |  ( x  e.  RR  /\  u  =  ( x  +  A ) ) }  =  { <. y ,  v >.  |  ( y  e.  RR  /\  v  =  ( y  +  A ) ) }
301, 21, 293eqtri 2307 . . . . . . . . . . 11  |-  F  =  { <. y ,  v
>.  |  ( y  e.  RR  /\  v  =  ( y  +  A
) ) }
31 df-mpt 4079 . . . . . . . . . . 11  |-  ( y  e.  RR  |->  ( y  +  A ) )  =  { <. y ,  v >.  |  ( y  e.  RR  /\  v  =  ( y  +  A ) ) }
3230, 31eqtr4i 2306 . . . . . . . . . 10  |-  F  =  ( y  e.  RR  |->  ( y  +  A
) )
3332fveq1i 5526 . . . . . . . . 9  |-  ( F `
 y )  =  ( ( y  e.  RR  |->  ( y  +  A ) ) `  y )
3419, 20, 333eqtr3g 2338 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
( x  e.  RR  |->  ( x  +  A
) ) `  x
)  =  ( ( y  e.  RR  |->  ( y  +  A ) ) `  y ) )
35 simprl 732 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
3635ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  x  e.  RR )
37 simplrl 736 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
x  +  A )  e.  RR )
38 eqid 2283 . . . . . . . . . 10  |-  ( x  e.  RR  |->  ( x  +  A ) )  =  ( x  e.  RR  |->  ( x  +  A ) )
3938fvmpt2 5608 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  ( x  +  A
)  e.  RR )  ->  ( ( x  e.  RR  |->  ( x  +  A ) ) `
 x )  =  ( x  +  A
) )
4036, 37, 39syl2anc 642 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
( x  e.  RR  |->  ( x  +  A
) ) `  x
)  =  ( x  +  A ) )
41 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  (
y  +  A )  e.  RR ) )  ->  y  e.  RR )
42 simprr 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  (
y  +  A )  e.  RR ) )  ->  ( y  +  A )  e.  RR )
4341, 42jca 518 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  (
y  +  A )  e.  RR ) )  ->  ( y  e.  RR  /\  ( y  +  A )  e.  RR ) )
4443adantr 451 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
y  e.  RR  /\  ( y  +  A
)  e.  RR ) )
45 eqid 2283 . . . . . . . . . 10  |-  ( y  e.  RR  |->  ( y  +  A ) )  =  ( y  e.  RR  |->  ( y  +  A ) )
4645fvmpt2 5608 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  ( y  +  A
)  e.  RR )  ->  ( ( y  e.  RR  |->  ( y  +  A ) ) `
 y )  =  ( y  +  A
) )
4744, 46syl 15 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
( y  e.  RR  |->  ( y  +  A
) ) `  y
)  =  ( y  +  A ) )
4834, 40, 473eqtr3d 2323 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
x  +  A )  =  ( y  +  A ) )
49 recn 8827 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
5049ad2antrl 708 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  CC )
51 recn 8827 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  y  e.  CC )
5251ad2antll 709 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  y  e.  CC )
53 recn 8827 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
5453adantr 451 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A  e.  CC )
5550, 52, 543jca 1132 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  e.  CC  /\  y  e.  CC  /\  A  e.  CC ) )
5655ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
x  e.  CC  /\  y  e.  CC  /\  A  e.  CC ) )
57 addcan2 8997 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  A  e.  CC )  ->  (
( x  +  A
)  =  ( y  +  A )  <->  x  =  y ) )
5856, 57syl 15 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  (
( x  +  A
)  =  ( y  +  A )  <->  x  =  y ) )
5948, 58mpbid 201 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( ( x  +  A )  e.  RR  /\  ( y  +  A
)  e.  RR ) )  /\  ( F `
 x )  =  ( F `  y
) )  ->  x  =  y )
6059exp31 587 . . . . 5  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( (
( x  +  A
)  e.  RR  /\  ( y  +  A
)  e.  RR )  ->  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
6118, 60mpd 14 . . . 4  |-  ( ( A  e.  RR  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
6261ralrimivva 2635 . . 3  |-  ( A  e.  RR  ->  A. x  e.  RR  A. y  e.  RR  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) )
63 dff13 5783 . . 3  |-  ( F : RR -1-1-> RR  <->  ( F : RR --> RR  /\  A. x  e.  RR  A. y  e.  RR  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
6412, 62, 63sylanbrc 645 . 2  |-  ( A  e.  RR  ->  F : RR -1-1-> RR )
65 df-fo 5261 . . 3  |-  ( F : RR -onto-> RR  <->  ( F  Fn  RR  /\  ran  F  =  RR ) )
666, 7, 65sylanbrc 645 . 2  |-  ( A  e.  RR  ->  F : RR -onto-> RR )
67 df-f1o 5262 . 2  |-  ( F : RR -1-1-onto-> RR  <->  ( F : RR
-1-1-> RR  /\  F : RR -onto-> RR ) )
6864, 66, 67sylanbrc 645 1  |-  ( A  e.  RR  ->  F : RR -1-1-onto-> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {copab 4076    e. cmpt 4077   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    + caddc 8740
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-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-po 4314  df-so 4315  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040
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