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Theorem dnnumch3 27144
Description: Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
dnnumch.a  |-  ( ph  ->  A  e.  V )
dnnumch.g  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
Assertion
Ref Expression
dnnumch3  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
Distinct variable groups:    x, F, y    x, G, y, z   
x, A, y, z    ph, x
Allowed substitution hints:    ph( y, z)    F( z)    V( x, y, z)

Proof of Theorem dnnumch3
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 5033 . . . . 5  |-  ( `' F " { x } )  C_  dom  F
2 dnnumch.f . . . . . . 7  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
32tfr1 6413 . . . . . 6  |-  F  Fn  On
4 fndm 5343 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 8 . . . . 5  |-  dom  F  =  On
61, 5sseqtri 3210 . . . 4  |-  ( `' F " { x } )  C_  On
7 dnnumch.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
8 dnnumch.g . . . . . . 7  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
92, 7, 8dnnumch2 27142 . . . . . 6  |-  ( ph  ->  A  C_  ran  F )
109sselda 3180 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  F )
11 inisegn0 27140 . . . . 5  |-  ( x  e.  ran  F  <->  ( `' F " { x }
)  =/=  (/) )
1210, 11sylib 188 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( `' F " { x } )  =/=  (/) )
13 oninton 4591 . . . 4  |-  ( ( ( `' F " { x } ) 
C_  On  /\  ( `' F " { x } )  =/=  (/) )  ->  |^| ( `' F " { x } )  e.  On )
146, 12, 13sylancr 644 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  |^| ( `' F " { x } )  e.  On )
15 eqid 2283 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
1614, 15fmptd 5684 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
172, 7, 8dnnumch3lem 27143 . . . . . 6  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1817adantrr 697 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
192, 7, 8dnnumch3lem 27143 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2019adantrl 696 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2118, 20eqeq12d 2297 . . . 4  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  <->  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) ) )
22 fveq2 5525 . . . . . . 7  |-  ( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  ( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
2322adantl 452 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
24 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' F " { v } )  C_  dom  F
2524, 5sseqtri 3210 . . . . . . . . . 10  |-  ( `' F " { v } )  C_  On
269sselda 3180 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  v  e.  ran  F )
27 inisegn0 27140 . . . . . . . . . . 11  |-  ( v  e.  ran  F  <->  ( `' F " { v } )  =/=  (/) )
2826, 27sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( `' F " { v } )  =/=  (/) )
29 onint 4586 . . . . . . . . . 10  |-  ( ( ( `' F " { v } ) 
C_  On  /\  ( `' F " { v } )  =/=  (/) )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
3025, 28, 29sylancr 644 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  A )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
31 fniniseg 5646 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  <-> 
( |^| ( `' F " { v } )  e.  On  /\  ( F `  |^| ( `' F " { v } ) )  =  v ) ) )
323, 31ax-mp 8 . . . . . . . . . 10  |-  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  <->  ( |^| ( `' F " { v } )  e.  On  /\  ( F `  |^| ( `' F " { v } ) )  =  v ) )
3332simprbi 450 . . . . . . . . 9  |-  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3430, 33syl 15 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3534adantrr 697 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
3635adantr 451 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
37 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' F " { w } )  C_  dom  F
3837, 5sseqtri 3210 . . . . . . . . . 10  |-  ( `' F " { w } )  C_  On
399sselda 3180 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
40 inisegn0 27140 . . . . . . . . . . 11  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
4139, 40sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
42 onint 4586 . . . . . . . . . 10  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
4338, 41, 42sylancr 644 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
44 fniniseg 5646 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  <-> 
( |^| ( `' F " { w } )  e.  On  /\  ( F `  |^| ( `' F " { w } ) )  =  w ) ) )
453, 44ax-mp 8 . . . . . . . . . 10  |-  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  <->  ( |^| ( `' F " { w } )  e.  On  /\  ( F `  |^| ( `' F " { w } ) )  =  w ) )
4645simprbi 450 . . . . . . . . 9  |-  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4743, 46syl 15 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4847adantrl 696 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
4948adantr 451 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
5023, 36, 493eqtr3d 2323 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
v  =  w )
5150ex 423 . . . 4  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  v  =  w ) )
5221, 51sylbid 206 . . 3  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  ->  v  =  w ) )
5352ralrimivva 2635 . 2  |-  ( ph  ->  A. v  e.  A  A. w  e.  A  ( ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  ->  v  =  w ) )
54 dff13 5783 . 2  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  <->  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On  /\  A. v  e.  A  A. w  e.  A  (
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  ->  v  =  w ) ) )
5516, 53, 54sylanbrc 645 1  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   |^|cint 3862    e. cmpt 4077   Oncon0 4392   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  dnwech  27145
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  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-recs 6388
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