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Theorem dnnumch3lem 27143
Description: Value of the ordinal injection 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
dnnumch3lem  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
Distinct variable groups:    w, F, x, y    w, G, x, y, z    w, A, x, y, z    ph, x, w
Allowed substitution hints:    ph( y, z)    F( z)    V( x, y, z, w)

Proof of Theorem dnnumch3lem
StepHypRef Expression
1 simpr 447 . 2  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
2 cnvimass 5033 . . . 4  |-  ( `' F " { w } )  C_  dom  F
3 dnnumch.f . . . . . 6  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
43tfr1 6413 . . . . 5  |-  F  Fn  On
5 fndm 5343 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 8 . . . 4  |-  dom  F  =  On
72, 6sseqtri 3210 . . 3  |-  ( `' F " { w } )  C_  On
8 dnnumch.a . . . . . 6  |-  ( ph  ->  A  e.  V )
9 dnnumch.g . . . . . 6  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
103, 8, 9dnnumch2 27142 . . . . 5  |-  ( ph  ->  A  C_  ran  F )
1110sselda 3180 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
12 inisegn0 27140 . . . 4  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
1311, 12sylib 188 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
14 oninton 4591 . . 3  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  On )
157, 13, 14sylancr 644 . 2  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  On )
16 sneq 3651 . . . . 5  |-  ( x  =  w  ->  { x }  =  { w } )
1716imaeq2d 5012 . . . 4  |-  ( x  =  w  ->  ( `' F " { x } )  =  ( `' F " { w } ) )
1817inteqd 3867 . . 3  |-  ( x  =  w  ->  |^| ( `' F " { x } )  =  |^| ( `' F " { w } ) )
19 eqid 2283 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
2018, 19fvmptg 5600 . 2  |-  ( ( w  e.  A  /\  |^| ( `' F " { w } )  e.  On )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
211, 15, 20syl2anc 642 1  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  dnnumch3  27144  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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