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Theorem dnnumch3 30625
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 5357 . . . . 5  |-  ( `' F " { x } )  C_  dom  F
2 dnnumch.f . . . . . . 7  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
32tfr1 7066 . . . . . 6  |-  F  Fn  On
4 fndm 5680 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 5 . . . . 5  |-  dom  F  =  On
61, 5sseqtri 3536 . . . 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 30623 . . . . . 6  |-  ( ph  ->  A  C_  ran  F )
109sselda 3504 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  F )
11 inisegn0 30621 . . . . 5  |-  ( x  e.  ran  F  <->  ( `' F " { x }
)  =/=  (/) )
1210, 11sylib 196 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( `' F " { x } )  =/=  (/) )
13 oninton 6619 . . . 4  |-  ( ( ( `' F " { x } ) 
C_  On  /\  ( `' F " { x } )  =/=  (/) )  ->  |^| ( `' F " { x } )  e.  On )
146, 12, 13sylancr 663 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  |^| ( `' F " { x } )  e.  On )
15 eqid 2467 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
1614, 15fmptd 6045 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
172, 7, 8dnnumch3lem 30624 . . . . . 6  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1817adantrr 716 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
192, 7, 8dnnumch3lem 30624 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2019adantrl 715 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2118, 20eqeq12d 2489 . . . 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 5866 . . . . . . 7  |-  ( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  ( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
2322adantl 466 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
24 cnvimass 5357 . . . . . . . . . . 11  |-  ( `' F " { v } )  C_  dom  F
2524, 5sseqtri 3536 . . . . . . . . . 10  |-  ( `' F " { v } )  C_  On
269sselda 3504 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  v  e.  ran  F )
27 inisegn0 30621 . . . . . . . . . . 11  |-  ( v  e.  ran  F  <->  ( `' F " { v } )  =/=  (/) )
2826, 27sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( `' F " { v } )  =/=  (/) )
29 onint 6614 . . . . . . . . . 10  |-  ( ( ( `' F " { v } ) 
C_  On  /\  ( `' F " { v } )  =/=  (/) )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
3025, 28, 29sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  A )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
31 fniniseg 6002 . . . . . . . . . . 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 464 . . . . . . . . 9  |-  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3430, 33syl 16 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3534adantrr 716 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
3635adantr 465 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
37 cnvimass 5357 . . . . . . . . . . 11  |-  ( `' F " { w } )  C_  dom  F
3837, 5sseqtri 3536 . . . . . . . . . 10  |-  ( `' F " { w } )  C_  On
399sselda 3504 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
40 inisegn0 30621 . . . . . . . . . . 11  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
4139, 40sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
42 onint 6614 . . . . . . . . . 10  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
4338, 41, 42sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
44 fniniseg 6002 . . . . . . . . . . 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 464 . . . . . . . . 9  |-  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4743, 46syl 16 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4847adantrl 715 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
4948adantr 465 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
5023, 36, 493eqtr3d 2516 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
v  =  w )
5150ex 434 . . . 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 2885 . 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 6154 . 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 664 1  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   |^|cint 4282    |-> cmpt 4505   Oncon0 4878   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   ` cfv 5588  recscrecs 7041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-recs 7042
This theorem is referenced by:  dnwech  30626
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