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Theorem dnnumch3 26815
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 5166 . . . . 5  |-  ( `' F " { x } )  C_  dom  F
2 dnnumch.f . . . . . . 7  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
32tfr1 6596 . . . . . 6  |-  F  Fn  On
4 fndm 5486 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 8 . . . . 5  |-  dom  F  =  On
61, 5sseqtri 3325 . . . 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 26813 . . . . . 6  |-  ( ph  ->  A  C_  ran  F )
109sselda 3293 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  F )
11 inisegn0 26811 . . . . 5  |-  ( x  e.  ran  F  <->  ( `' F " { x }
)  =/=  (/) )
1210, 11sylib 189 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( `' F " { x } )  =/=  (/) )
13 oninton 4722 . . . 4  |-  ( ( ( `' F " { x } ) 
C_  On  /\  ( `' F " { x } )  =/=  (/) )  ->  |^| ( `' F " { x } )  e.  On )
146, 12, 13sylancr 645 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  |^| ( `' F " { x } )  e.  On )
15 eqid 2389 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
1614, 15fmptd 5834 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
172, 7, 8dnnumch3lem 26814 . . . . . 6  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1817adantrr 698 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
192, 7, 8dnnumch3lem 26814 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2019adantrl 697 . . . . 5  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2118, 20eqeq12d 2403 . . . 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 5670 . . . . . . 7  |-  ( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  ( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
2322adantl 453 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  ( F `  |^| ( `' F " { w } ) ) )
24 cnvimass 5166 . . . . . . . . . . 11  |-  ( `' F " { v } )  C_  dom  F
2524, 5sseqtri 3325 . . . . . . . . . 10  |-  ( `' F " { v } )  C_  On
269sselda 3293 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  v  e.  ran  F )
27 inisegn0 26811 . . . . . . . . . . 11  |-  ( v  e.  ran  F  <->  ( `' F " { v } )  =/=  (/) )
2826, 27sylib 189 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( `' F " { v } )  =/=  (/) )
29 onint 4717 . . . . . . . . . 10  |-  ( ( ( `' F " { v } ) 
C_  On  /\  ( `' F " { v } )  =/=  (/) )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
3025, 28, 29sylancr 645 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  A )  ->  |^| ( `' F " { v } )  e.  ( `' F " { v } ) )
31 fniniseg 5792 . . . . . . . . . . 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 451 . . . . . . . . 9  |-  ( |^| ( `' F " { v } )  e.  ( `' F " { v } )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3430, 33syl 16 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  |^| ( `' F " { v } ) )  =  v )
3534adantrr 698 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
3635adantr 452 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { v } ) )  =  v )
37 cnvimass 5166 . . . . . . . . . . 11  |-  ( `' F " { w } )  C_  dom  F
3837, 5sseqtri 3325 . . . . . . . . . 10  |-  ( `' F " { w } )  C_  On
399sselda 3293 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
40 inisegn0 26811 . . . . . . . . . . 11  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
4139, 40sylib 189 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
42 onint 4717 . . . . . . . . . 10  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
4338, 41, 42sylancr 645 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  ( `' F " { w } ) )
44 fniniseg 5792 . . . . . . . . . . 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 451 . . . . . . . . 9  |-  ( |^| ( `' F " { w } )  e.  ( `' F " { w } )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4743, 46syl 16 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  |^| ( `' F " { w } ) )  =  w )
4847adantrl 697 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
4948adantr 452 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
( F `  |^| ( `' F " { w } ) )  =  w )
5023, 36, 493eqtr3d 2429 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  A  /\  w  e.  A )
)  /\  |^| ( `' F " { v } )  =  |^| ( `' F " { w } ) )  -> 
v  =  w )
5150ex 424 . . . 4  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( |^| ( `' F " { v } )  =  |^| ( `' F " { w } )  ->  v  =  w ) )
5221, 51sylbid 207 . . 3  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  ->  v  =  w ) )
5352ralrimivva 2743 . 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 5945 . 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 646 1  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   _Vcvv 2901    \ cdif 3262    C_ wss 3265   (/)c0 3573   ~Pcpw 3744   {csn 3759   |^|cint 3994    e. cmpt 4209   Oncon0 4524   `'ccnv 4819   dom cdm 4820   ran crn 4821   "cima 4823    Fn wfn 5391   -->wf 5392   -1-1->wf1 5393   ` cfv 5396  recscrecs 6570
This theorem is referenced by:  dnwech  26816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-suc 4530  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-recs 6571
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