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Theorem dnnumch3 31235
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 5345 . . . . 5  |-  ( `' F " { x } )  C_  dom  F
2 dnnumch.f . . . . . . 7  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
32tfr1 7058 . . . . . 6  |-  F  Fn  On
4 fndm 5662 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 5 . . . . 5  |-  dom  F  =  On
61, 5sseqtri 3521 . . . 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 31233 . . . . . 6  |-  ( ph  ->  A  C_  ran  F )
109sselda 3489 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  F )
11 inisegn0 31231 . . . . 5  |-  ( x  e.  ran  F  <->  ( `' F " { x }
)  =/=  (/) )
1210, 11sylib 196 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( `' F " { x } )  =/=  (/) )
13 oninton 6608 . . . 4  |-  ( ( ( `' F " { x } ) 
C_  On  /\  ( `' F " { x } )  =/=  (/) )  ->  |^| ( `' F " { x } )  e.  On )
146, 12, 13sylancr 661 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  |^| ( `' F " { x } )  e.  On )
15 eqid 2454 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
1614, 15fmptd 6031 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
172, 7, 8dnnumch3lem 31234 . . . . . 6  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1817adantrr 714 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
192, 7, 8dnnumch3lem 31234 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2019adantrl 713 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2118, 20eqeq12d 2476 . . . 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 5848 . . . . . . 7  |-  ( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  ( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
2322adantl 464 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
24 cnvimass 5345 . . . . . . . . . . 11  |-  ( `' F " { v } )  C_  dom  F
2524, 5sseqtri 3521 . . . . . . . . . 10  |-  ( `' F " { v } )  C_  On
269sselda 3489 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  v  e.  ran  F )
27 inisegn0 31231 . . . . . . . . . . 11  |-  ( v  e.  ran  F  <->  ( `' F " { v } )  =/=  (/) )
2826, 27sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( `' F " { v } )  =/=  (/) )
29 onint 6603 . . . . . . . . . 10  |-  ( ( ( `' F " { v } ) 
C_  On  /\  ( `' F " { v } )  =/=  (/) )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
3025, 28, 29sylancr 661 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  A )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
31 fniniseg 5984 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  <-> 
( |^| ( `' F " { v } )  e.  On  /\  ( F `  |^| ( `' F " { v } ) )  =  v ) ) )
323, 31ax-mp 5 . . . . . . . . . 10  |-  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  <->  ( |^| ( `' F " { v } )  e.  On  /\  ( F `  |^| ( `' F " { v } ) )  =  v ) )
3332simprbi 462 . . . . . . . . 9  |-  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3430, 33syl 16 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3534adantrr 714 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
3635adantr 463 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
37 cnvimass 5345 . . . . . . . . . . 11  |-  ( `' F " { w } )  C_  dom  F
3837, 5sseqtri 3521 . . . . . . . . . 10  |-  ( `' F " { w } )  C_  On
399sselda 3489 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
40 inisegn0 31231 . . . . . . . . . . 11  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
4139, 40sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
42 onint 6603 . . . . . . . . . 10  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
4338, 41, 42sylancr 661 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
44 fniniseg 5984 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  <-> 
( |^| ( `' F " { w } )  e.  On  /\  ( F `  |^| ( `' F " { w } ) )  =  w ) ) )
453, 44ax-mp 5 . . . . . . . . . 10  |-  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  <->  ( |^| ( `' F " { w } )  e.  On  /\  ( F `  |^| ( `' F " { w } ) )  =  w ) )
4645simprbi 462 . . . . . . . . 9  |-  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4743, 46syl 16 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4847adantrl 713 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
4948adantr 463 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
5023, 36, 493eqtr3d 2503 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
v  =  w )
5150ex 432 . . . 4  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  v  =  w ) )
5221, 51sylbid 215 . . 3  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  ->  v  =  w ) )
5352ralrimivva 2875 . 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 6141 . 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 662 1  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   |^|cint 4271    |-> cmpt 4497   Oncon0 4867   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   -1-1->wf1 5567   ` cfv 5570  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-recs 7034
This theorem is referenced by:  dnwech  31236
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