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Theorem aomclem3 26822
Description: Lemma for dfac11 26829. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
aomclem3.b  |-  B  =  { <. a ,  b
>.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a
)  /\  A. d  e.  ( R1 `  U. dom  z ) ( d ( z `  U. dom  z ) c  -> 
( d  e.  a  <-> 
d  e.  b ) ) ) }
aomclem3.c  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
aomclem3.d  |-  D  = recs ( ( a  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  a ) ) ) )
aomclem3.e  |-  E  =  { <. a ,  b
>.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }
aomclem3.on  |-  ( ph  ->  dom  z  e.  On )
aomclem3.su  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
aomclem3.we  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
aomclem3.a  |-  ( ph  ->  A  e.  On )
aomclem3.za  |-  ( ph  ->  dom  z  C_  A
)
aomclem3.y  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
Assertion
Ref Expression
aomclem3  |-  ( ph  ->  E  We  ( R1
`  dom  z )
)
Distinct variable groups:    y, z,
a, b, c, d    ph, a, b    C, a, b, c, d    D, a, b, c, d
Allowed substitution hints:    ph( y, z, c, d)    A( y, z, a, b, c, d)    B( y, z, a, b, c, d)    C( y, z)    D( y, z)    E( y, z, a, b, c, d)

Proof of Theorem aomclem3
StepHypRef Expression
1 aomclem3.d . . 3  |-  D  = recs ( ( a  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  a ) ) ) )
2 rneq 5035 . . . . . . 7  |-  ( a  =  c  ->  ran  a  =  ran  c )
32difeq2d 3408 . . . . . 6  |-  ( a  =  c  ->  (
( R1 `  dom  z )  \  ran  a )  =  ( ( R1 `  dom  z )  \  ran  c ) )
43fveq2d 5672 . . . . 5  |-  ( a  =  c  ->  ( C `  ( ( R1 `  dom  z ) 
\  ran  a )
)  =  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
54cbvmptv 4241 . . . 4  |-  ( a  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  a ) ) )  =  ( c  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
6 recseq 6570 . . . 4  |-  ( ( a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
)  \  ran  a ) ) )  =  ( c  e.  _V  |->  ( C `  ( ( R1 `  dom  z
)  \  ran  c ) ) )  -> recs ( ( a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
)  \  ran  a ) ) ) )  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) ) )
75, 6ax-mp 8 . . 3  |- recs ( ( a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
)  \  ran  a ) ) ) )  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) )
81, 7eqtri 2407 . 2  |-  D  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) )
9 fvex 5682 . . 3  |-  ( R1
`  dom  z )  e.  _V
109a1i 11 . 2  |-  ( ph  ->  ( R1 `  dom  z )  e.  _V )
11 aomclem3.b . . . 4  |-  B  =  { <. a ,  b
>.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a
)  /\  A. d  e.  ( R1 `  U. dom  z ) ( d ( z `  U. dom  z ) c  -> 
( d  e.  a  <-> 
d  e.  b ) ) ) }
12 aomclem3.c . . . 4  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
13 aomclem3.on . . . 4  |-  ( ph  ->  dom  z  e.  On )
14 aomclem3.su . . . 4  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
15 aomclem3.we . . . 4  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
16 aomclem3.a . . . 4  |-  ( ph  ->  A  e.  On )
17 aomclem3.za . . . 4  |-  ( ph  ->  dom  z  C_  A
)
18 aomclem3.y . . . 4  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
1911, 12, 13, 14, 15, 16, 17, 18aomclem2 26821 . . 3  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
20 neeq1 2558 . . . . 5  |-  ( a  =  d  ->  (
a  =/=  (/)  <->  d  =/=  (/) ) )
21 fveq2 5668 . . . . . 6  |-  ( a  =  d  ->  ( C `  a )  =  ( C `  d ) )
22 id 20 . . . . . 6  |-  ( a  =  d  ->  a  =  d )
2321, 22eleq12d 2455 . . . . 5  |-  ( a  =  d  ->  (
( C `  a
)  e.  a  <->  ( C `  d )  e.  d ) )
2420, 23imbi12d 312 . . . 4  |-  ( a  =  d  ->  (
( a  =/=  (/)  ->  ( C `  a )  e.  a )  <->  ( d  =/=  (/)  ->  ( C `  d )  e.  d ) ) )
2524cbvralv 2875 . . 3  |-  ( A. a  e.  ~P  ( R1 `  dom  z ) ( a  =/=  (/)  ->  ( C `  a )  e.  a )  <->  A. d  e.  ~P  ( R1 `  dom  z ) ( d  =/=  (/)  ->  ( C `  d )  e.  d ) )
2619, 25sylib 189 . 2  |-  ( ph  ->  A. d  e.  ~P  ( R1 `  dom  z
) ( d  =/=  (/)  ->  ( C `  d )  e.  d ) )
27 aomclem3.e . 2  |-  E  =  { <. a ,  b
>.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }
288, 10, 26, 27dnwech 26814 1  |-  ( ph  ->  E  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   _Vcvv 2899    \ cdif 3260    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   {csn 3757   U.cuni 3957   |^|cint 3992   class class class wbr 4153   {copab 4206    e. cmpt 4207    We wwe 4481   Oncon0 4522   suc csuc 4524   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821   ` cfv 5394  recscrecs 6568   Fincfn 7045   supcsup 7380   R1cr1 7621
This theorem is referenced by:  aomclem5  26824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-er 6841  df-map 6956  df-en 7046  df-fin 7049  df-sup 7381  df-r1 7623
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