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Theorem dnnumch3lem 27122
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 449 . 2  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
2 cnvimass 5225 . . . 4  |-  ( `' F " { w } )  C_  dom  F
3 dnnumch.f . . . . . 6  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
43tfr1 6659 . . . . 5  |-  F  Fn  On
5 fndm 5545 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 8 . . . 4  |-  dom  F  =  On
72, 6sseqtri 3381 . . 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 27121 . . . . 5  |-  ( ph  ->  A  C_  ran  F )
1110sselda 3349 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
12 inisegn0 27119 . . . 4  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
1311, 12sylib 190 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
14 oninton 4781 . . 3  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  On )
157, 13, 14sylancr 646 . 2  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  On )
16 sneq 3826 . . . . 5  |-  ( x  =  w  ->  { x }  =  { w } )
1716imaeq2d 5204 . . . 4  |-  ( x  =  w  ->  ( `' F " { x } )  =  ( `' F " { w } ) )
1817inteqd 4056 . . 3  |-  ( x  =  w  ->  |^| ( `' F " { x } )  =  |^| ( `' F " { w } ) )
19 eqid 2437 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
2018, 19fvmptg 5805 . 2  |-  ( ( w  e.  A  /\  |^| ( `' F " { w } )  e.  On )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
211, 15, 20syl2anc 644 1  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   _Vcvv 2957    \ cdif 3318    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   {csn 3815   |^|cint 4051    e. cmpt 4267   Oncon0 4582   `'ccnv 4878   dom cdm 4879   ran crn 4880   "cima 4882    Fn wfn 5450   ` cfv 5455  recscrecs 6633
This theorem is referenced by:  dnnumch3  27123  dnwech  27124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-recs 6634
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