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Theorem aomclem3 27112
Description: Lemma for dfac11 27118. 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 5087 . . . . . . 7  |-  ( a  =  c  ->  ran  a  =  ran  c )
32difeq2d 3457 . . . . . 6  |-  ( a  =  c  ->  (
( R1 `  dom  z )  \  ran  a )  =  ( ( R1 `  dom  z )  \  ran  c ) )
43fveq2d 5724 . . . . 5  |-  ( a  =  c  ->  ( C `  ( ( R1 `  dom  z ) 
\  ran  a )
)  =  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
54cbvmptv 4292 . . . 4  |-  ( a  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  a ) ) )  =  ( c  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
6 recseq 6626 . . . 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 2455 . 2  |-  D  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) )
9 fvex 5734 . . 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 27111 . . 3  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
20 neeq1 2606 . . . . 5  |-  ( a  =  d  ->  (
a  =/=  (/)  <->  d  =/=  (/) ) )
21 fveq2 5720 . . . . . 6  |-  ( a  =  d  ->  ( C `  a )  =  ( C `  d ) )
22 id 20 . . . . . 6  |-  ( a  =  d  ->  a  =  d )
2321, 22eleq12d 2503 . . . . 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 2924 . . 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 27104 1  |-  ( ph  ->  E  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007   |^|cint 4042   class class class wbr 4204   {copab 4257    e. cmpt 4258    We wwe 4532   Oncon0 4573   suc csuc 4575   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873   ` cfv 5446  recscrecs 6624   Fincfn 7101   supcsup 7437   R1cr1 7680
This theorem is referenced by:  aomclem5  27114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-er 6897  df-map 7012  df-en 7102  df-fin 7105  df-sup 7438  df-r1 7682
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