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Theorem aomclem4 27257
Description: Lemma for dfac11 27263. Limit case. Patch together well-orderings constructed so far using fnwe2 27253 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem4.f  |-  F  =  { <. a ,  b
>.  |  ( ( rank `  a )  _E  ( rank `  b
)  \/  ( (
rank `  a )  =  ( rank `  b
)  /\  a (
z `  suc  ( rank `  a ) ) b ) ) }
aomclem4.on  |-  ( ph  ->  dom  z  e.  On )
aomclem4.su  |-  ( ph  ->  dom  z  =  U. dom  z )
aomclem4.we  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
Assertion
Ref Expression
aomclem4  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Distinct variable groups:    z, a,
b    ph, a, b
Allowed substitution hints:    ph( z)    F( z, a, b)

Proof of Theorem aomclem4
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 suceq 4473 . . 3  |-  ( c  =  ( rank `  a
)  ->  suc  c  =  suc  ( rank `  a
) )
21fveq2d 5545 . 2  |-  ( c  =  ( rank `  a
)  ->  ( z `  suc  c )  =  ( z `  suc  ( rank `  a )
) )
3 aomclem4.f . 2  |-  F  =  { <. a ,  b
>.  |  ( ( rank `  a )  _E  ( rank `  b
)  \/  ( (
rank `  a )  =  ( rank `  b
)  /\  a (
z `  suc  ( rank `  a ) ) b ) ) }
4 r1fnon 7455 . . . . . . . . . . . . . 14  |-  R1  Fn  On
5 fnfun 5357 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  R1
7 fndm 5359 . . . . . . . . . . . . . . 15  |-  ( R1  Fn  On  ->  dom  R1  =  On )
84, 7ax-mp 8 . . . . . . . . . . . . . 14  |-  dom  R1  =  On
98eqimss2i 3246 . . . . . . . . . . . . 13  |-  On  C_  dom  R1
106, 9pm3.2i 441 . . . . . . . . . . . 12  |-  ( Fun 
R1  /\  On  C_  dom  R1 )
11 aomclem4.on . . . . . . . . . . . 12  |-  ( ph  ->  dom  z  e.  On )
12 funfvima2 5770 . . . . . . . . . . . 12  |-  ( ( Fun  R1  /\  On  C_ 
dom  R1 )  ->  ( dom  z  e.  On  ->  ( R1 `  dom  z )  e.  ( R1 " On ) ) )
1310, 11, 12mpsyl 59 . . . . . . . . . . 11  |-  ( ph  ->  ( R1 `  dom  z )  e.  ( R1 " On ) )
14 elssuni 3871 . . . . . . . . . . 11  |-  ( ( R1 `  dom  z
)  e.  ( R1
" On )  -> 
( R1 `  dom  z )  C_  U. ( R1 " On ) )
1513, 14syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( R1 `  dom  z )  C_  U. ( R1 " On ) )
1615sselda 3193 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  U. ( R1 " On ) )
17 rankidb 7488 . . . . . . . . 9  |-  ( b  e.  U. ( R1
" On )  -> 
b  e.  ( R1
`  suc  ( rank `  b ) ) )
1816, 17syl 15 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  ( R1 `  suc  ( rank `  b )
) )
19 suceq 4473 . . . . . . . . . 10  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  suc  ( rank `  b )  =  suc  ( rank `  a )
)
2019fveq2d 5545 . . . . . . . . 9  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( R1 ` 
suc  ( rank `  b
) )  =  ( R1 `  suc  ( rank `  a ) ) )
2120eleq2d 2363 . . . . . . . 8  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( b  e.  ( R1 `  suc  ( rank `  b )
)  <->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2218, 21syl5ibcom 211 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  (
( rank `  b )  =  ( rank `  a
)  ->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2322expimpd 586 . . . . . 6  |-  ( ph  ->  ( ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) )  ->  b  e.  ( R1 `  suc  ( rank `  a )
) ) )
2423ss2abdv 3259 . . . . 5  |-  ( ph  ->  { b  |  ( b  e.  ( R1
`  dom  z )  /\  ( rank `  b
)  =  ( rank `  a ) ) } 
C_  { b  |  b  e.  ( R1
`  suc  ( rank `  a ) ) } )
25 df-rab 2565 . . . . 5  |-  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  =  {
b  |  ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) ) }
26 abid2 2413 . . . . . 6  |-  { b  |  b  e.  ( R1 `  suc  ( rank `  a ) ) }  =  ( R1
`  suc  ( rank `  a ) )
2726eqcomi 2300 . . . . 5  |-  ( R1
`  suc  ( rank `  a ) )  =  { b  |  b  e.  ( R1 `  suc  ( rank `  a
) ) }
2824, 25, 273sstr4g 3232 . . . 4  |-  ( ph  ->  { b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) }  C_  ( R1 `  suc  ( rank `  a
) ) )
2928adantr 451 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  C_  ( R1 `  suc  ( rank `  a ) ) )
30 rankr1ai 7486 . . . . . 6  |-  ( a  e.  ( R1 `  dom  z )  ->  ( rank `  a )  e. 
dom  z )
3130adantl 452 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  ( rank `  a )  e. 
dom  z )
32 eloni 4418 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
3311, 32syl 15 . . . . . . 7  |-  ( ph  ->  Ord  dom  z )
34 aomclem4.su . . . . . . 7  |-  ( ph  ->  dom  z  =  U. dom  z )
35 limsuc2 27240 . . . . . . 7  |-  ( ( Ord  dom  z  /\  dom  z  =  U. dom  z )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3633, 34, 35syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( rank `  a
)  e.  dom  z  <->  suc  ( rank `  a
)  e.  dom  z
) )
3736adantr 451 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3831, 37mpbid 201 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  suc  ( rank `  a )  e.  dom  z )
39 aomclem4.we . . . . . 6  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
40 fveq2 5541 . . . . . . . 8  |-  ( a  =  b  ->  (
z `  a )  =  ( z `  b ) )
41 fveq2 5541 . . . . . . . 8  |-  ( a  =  b  ->  ( R1 `  a )  =  ( R1 `  b
) )
4240, 41weeq12d 27239 . . . . . . 7  |-  ( a  =  b  ->  (
( z `  a
)  We  ( R1
`  a )  <->  ( z `  b )  We  ( R1 `  b ) ) )
4342cbvralv 2777 . . . . . 6  |-  ( A. a  e.  dom  z ( z `  a )  We  ( R1 `  a )  <->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
4439, 43sylib 188 . . . . 5  |-  ( ph  ->  A. b  e.  dom  z ( z `  b )  We  ( R1 `  b ) )
4544adantr 451 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
46 fveq2 5541 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  (
z `  b )  =  ( z `  suc  ( rank `  a
) ) )
47 fveq2 5541 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  ( R1 `  b )  =  ( R1 `  suc  ( rank `  a )
) )
4846, 47weeq12d 27239 . . . . 5  |-  ( b  =  suc  ( rank `  a )  ->  (
( z `  b
)  We  ( R1
`  b )  <->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) ) )
4948rspcva 2895 . . . 4  |-  ( ( suc  ( rank `  a
)  e.  dom  z  /\  A. b  e.  dom  z ( z `  b )  We  ( R1 `  b ) )  ->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) )
5038, 45, 49syl2anc 642 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
) )
51 wess 4396 . . 3  |-  ( { b  e.  ( R1
`  dom  z )  |  ( rank `  b
)  =  ( rank `  a ) }  C_  ( R1 `  suc  ( rank `  a ) )  ->  ( ( z `
 suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
)  ->  ( z `  suc  ( rank `  a
) )  We  {
b  e.  ( R1
`  dom  z )  |  ( rank `  b
)  =  ( rank `  a ) } ) )
5229, 50, 51sylc 56 . 2  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We 
{ b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) } )
53 rankf 7482 . . . 4  |-  rank : U. ( R1 " On ) --> On
5453a1i 10 . . 3  |-  ( ph  ->  rank : U. ( R1 " On ) --> On )
55 fssres 5424 . . 3  |-  ( (
rank : U. ( R1
" On ) --> On 
/\  ( R1 `  dom  z )  C_  U. ( R1 " On ) )  ->  ( rank  |`  ( R1 `  dom  z ) ) : ( R1
`  dom  z ) --> On )
5654, 15, 55syl2anc 642 . 2  |-  ( ph  ->  ( rank  |`  ( R1
`  dom  z )
) : ( R1
`  dom  z ) --> On )
57 epweon 4591 . . 3  |-  _E  We  On
5857a1i 10 . 2  |-  ( ph  ->  _E  We  On )
592, 3, 52, 56, 58fnwe2 27253 1  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560    C_ wss 3165   U.cuni 3843   class class class wbr 4039   {copab 4092    _E cep 4319    We wwe 4367   Ord word 4407   Oncon0 4408   suc csuc 4410   dom cdm 4705    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271   R1cr1 7450   rankcrnk 7451
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-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-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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