Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aomclem3 Structured version   Unicode version

Theorem aomclem3 31244
Description: Lemma for dfac11 31250. 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 5217 . . . . . . 7  |-  ( a  =  c  ->  ran  a  =  ran  c )
32difeq2d 3608 . . . . . 6  |-  ( a  =  c  ->  (
( R1 `  dom  z )  \  ran  a )  =  ( ( R1 `  dom  z )  \  ran  c ) )
43fveq2d 5852 . . . . 5  |-  ( a  =  c  ->  ( C `  ( ( R1 `  dom  z ) 
\  ran  a )
)  =  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
54cbvmptv 4530 . . . 4  |-  ( a  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  a ) ) )  =  ( c  e.  _V  |->  ( C `
 ( ( R1
`  dom  z )  \  ran  c ) ) )
6 recseq 7035 . . . 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 5 . . 3  |- recs ( ( a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
)  \  ran  a ) ) ) )  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) )
81, 7eqtri 2483 . 2  |-  D  = recs ( ( c  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  c ) ) ) )
9 fvex 5858 . . 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 31243 . . 3  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
20 neeq1 2735 . . . . 5  |-  ( a  =  d  ->  (
a  =/=  (/)  <->  d  =/=  (/) ) )
21 fveq2 5848 . . . . . 6  |-  ( a  =  d  ->  ( C `  a )  =  ( C `  d ) )
22 id 22 . . . . . 6  |-  ( a  =  d  ->  a  =  d )
2321, 22eleq12d 2536 . . . . 5  |-  ( a  =  d  ->  (
( C `  a
)  e.  a  <->  ( C `  d )  e.  d ) )
2420, 23imbi12d 318 . . . 4  |-  ( a  =  d  ->  (
( a  =/=  (/)  ->  ( C `  a )  e.  a )  <->  ( d  =/=  (/)  ->  ( C `  d )  e.  d ) ) )
2524cbvralv 3081 . . 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 196 . 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 31236 1  |-  ( ph  ->  E  We  ( R1
`  dom  z )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235   |^|cint 4271   class class class wbr 4439   {copab 4496    |-> cmpt 4497    We wwe 4826   Oncon0 4867   suc csuc 4869   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   ` cfv 5570  recscrecs 7033   Fincfn 7509   supcsup 7892   R1cr1 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-er 7303  df-map 7414  df-en 7510  df-fin 7513  df-sup 7893  df-r1 8173
This theorem is referenced by:  aomclem5  31246
  Copyright terms: Public domain W3C validator