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Theorem dnnumch3 27247
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 5049 . . . . 5  |-  ( `' F " { x } )  C_  dom  F
2 dnnumch.f . . . . . . 7  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
32tfr1 6429 . . . . . 6  |-  F  Fn  On
4 fndm 5359 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 8 . . . . 5  |-  dom  F  =  On
61, 5sseqtri 3223 . . . 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 27245 . . . . . 6  |-  ( ph  ->  A  C_  ran  F )
109sselda 3193 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  ran  F )
11 inisegn0 27243 . . . . 5  |-  ( x  e.  ran  F  <->  ( `' F " { x }
)  =/=  (/) )
1210, 11sylib 188 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( `' F " { x } )  =/=  (/) )
13 oninton 4607 . . . 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 2296 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
1614, 15fmptd 5700 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
172, 7, 8dnnumch3lem 27246 . . . . . 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 27246 . . . . . 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 2310 . . . 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 5541 . . . . . . 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 5049 . . . . . . . . . . 11  |-  ( `' F " { v } )  C_  dom  F
2524, 5sseqtri 3223 . . . . . . . . . 10  |-  ( `' F " { v } )  C_  On
269sselda 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  v  e.  ran  F )
27 inisegn0 27243 . . . . . . . . . . 11  |-  ( v  e.  ran  F  <->  ( `' F " { v } )  =/=  (/) )
2826, 27sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( `' F " { v } )  =/=  (/) )
29 onint 4602 . . . . . . . . . 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 5662 . . . . . . . . . . 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 5049 . . . . . . . . . . 11  |-  ( `' F " { w } )  C_  dom  F
3837, 5sseqtri 3223 . . . . . . . . . 10  |-  ( `' F " { w } )  C_  On
399sselda 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
40 inisegn0 27243 . . . . . . . . . . 11  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
4139, 40sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
42 onint 4602 . . . . . . . . . 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 5662 . . . . . . . . . . 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 2336 . . . . 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 2648 . 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 5799 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   |^|cint 3878    e. cmpt 4093   Oncon0 4408   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   ` cfv 5271  recscrecs 6403
This theorem is referenced by:  dnwech  27248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404
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