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Theorem dnnumch3lem 29422
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 461 . 2  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
2 cnvimass 5208 . . . 4  |-  ( `' F " { w } )  C_  dom  F
3 dnnumch.f . . . . . 6  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
43tfr1 6875 . . . . 5  |-  F  Fn  On
5 fndm 5529 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 5 . . . 4  |-  dom  F  =  On
72, 6sseqtri 3407 . . 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 29421 . . . . 5  |-  ( ph  ->  A  C_  ran  F )
1110sselda 3375 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
12 inisegn0 29419 . . . 4  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
1311, 12sylib 196 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
14 oninton 6430 . . 3  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  On )
157, 13, 14sylancr 663 . 2  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  On )
16 sneq 3906 . . . . 5  |-  ( x  =  w  ->  { x }  =  { w } )
1716imaeq2d 5188 . . . 4  |-  ( x  =  w  ->  ( `' F " { x } )  =  ( `' F " { w } ) )
1817inteqd 4152 . . 3  |-  ( x  =  w  ->  |^| ( `' F " { x } )  =  |^| ( `' F " { w } ) )
19 eqid 2443 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
2018, 19fvmptg 5791 . 2  |-  ( ( w  e.  A  /\  |^| ( `' F " { w } )  e.  On )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
211, 15, 20syl2anc 661 1  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2734   _Vcvv 2991    \ cdif 3344    C_ wss 3347   (/)c0 3656   ~Pcpw 3879   {csn 3896   |^|cint 4147    e. cmpt 4369   Oncon0 4738   `'ccnv 4858   dom cdm 4859   ran crn 4860   "cima 4862    Fn wfn 5432   ` cfv 5437  recscrecs 6850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-recs 6851
This theorem is referenced by:  dnnumch3  29423  dnwech  29424
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