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Theorem dnnumch3lem 35904
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 463 . 2  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
2 cnvimass 5188 . . . 4  |-  ( `' F " { w } )  C_  dom  F
3 dnnumch.f . . . . . 6  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
43tfr1 7115 . . . . 5  |-  F  Fn  On
5 fndm 5675 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 5 . . . 4  |-  dom  F  =  On
72, 6sseqtri 3464 . . 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 35903 . . . . 5  |-  ( ph  ->  A  C_  ran  F )
1110sselda 3432 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  F )
12 inisegn0 5200 . . . 4  |-  ( w  e.  ran  F  <->  ( `' F " { w }
)  =/=  (/) )
1311, 12sylib 200 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( `' F " { w } )  =/=  (/) )
14 oninton 6627 . . 3  |-  ( ( ( `' F " { w } ) 
C_  On  /\  ( `' F " { w } )  =/=  (/) )  ->  |^| ( `' F " { w } )  e.  On )
157, 13, 14sylancr 669 . 2  |-  ( (
ph  /\  w  e.  A )  ->  |^| ( `' F " { w } )  e.  On )
16 sneq 3978 . . . . 5  |-  ( x  =  w  ->  { x }  =  { w } )
1716imaeq2d 5168 . . . 4  |-  ( x  =  w  ->  ( `' F " { x } )  =  ( `' F " { w } ) )
1817inteqd 4239 . . 3  |-  ( x  =  w  ->  |^| ( `' F " { x } )  =  |^| ( `' F " { w } ) )
19 eqid 2451 . . 3  |-  ( x  e.  A  |->  |^| ( `' F " { x } ) )  =  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )
2018, 19fvmptg 5946 . 2  |-  ( ( w  e.  A  /\  |^| ( `' F " { w } )  e.  On )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
211, 15, 20syl2anc 667 1  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   _Vcvv 3045    \ cdif 3401    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   |^|cint 4234    |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Oncon0 5423    Fn wfn 5577   ` cfv 5582  recscrecs 7089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-wrecs 7028  df-recs 7090
This theorem is referenced by:  dnnumch3  35905  dnwech  35906
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