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Theorem aomclem3 27256
Description: Lemma for dfac11 27263. 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 4920 . . . . . . 7  |-  ( a  =  c  ->  ran  a  =  ran  c )
32difeq2d 3307 . . . . . 6  |-  ( a  =  c  ->  (
( R1 `  dom  z )  \  ran  a )  =  ( ( R1 `  dom  z )  \  ran  c ) )
43fveq2d 5545 . . . . 5  |-  ( a  =  c  ->  ( C `  ( ( R1 `  dom  z ) 
\  ran  a )
)  =  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
54cbvmptv 4127 . . . 4  |-  ( a  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  a ) ) )  =  ( c  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
6 recseq 6405 . . . 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 2316 . 2  |-  D  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) )
9 fvex 5555 . . 3  |-  ( R1
`  dom  z )  e.  _V
109a1i 10 . 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 27255 . . 3  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
20 neeq1 2467 . . . . 5  |-  ( a  =  d  ->  (
a  =/=  (/)  <->  d  =/=  (/) ) )
21 fveq2 5541 . . . . . 6  |-  ( a  =  d  ->  ( C `  a )  =  ( C `  d ) )
22 id 19 . . . . . 6  |-  ( a  =  d  ->  a  =  d )
2321, 22eleq12d 2364 . . . . 5  |-  ( a  =  d  ->  (
( C `  a
)  e.  a  <->  ( C `  d )  e.  d ) )
2420, 23imbi12d 311 . . . 4  |-  ( a  =  d  ->  (
( a  =/=  (/)  ->  ( C `  a )  e.  a )  <->  ( d  =/=  (/)  ->  ( C `  d )  e.  d ) ) )
2524cbvralv 2777 . . 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 188 . 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 27248 1  |-  ( ph  ->  E  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843   |^|cint 3878   class class class wbr 4039   {copab 4092    e. cmpt 4093    We wwe 4367   Oncon0 4408   suc csuc 4410   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   ` cfv 5271  recscrecs 6403   Fincfn 6879   supcsup 7209   R1cr1 7450
This theorem is referenced by:  aomclem5  27258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-sup 7210  df-r1 7452
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