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Theorem aomclem2 27163
Description: Lemma for dfac11 27171. Successor case 2, a choice function for subsets of  ( R1 `  dom  z ). (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
aomclem2.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 ) ) ) }
aomclem2.c  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
aomclem2.on  |-  ( ph  ->  dom  z  e.  On )
aomclem2.su  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
aomclem2.we  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
aomclem2.a  |-  ( ph  ->  A  e.  On )
aomclem2.za  |-  ( ph  ->  dom  z  C_  A
)
aomclem2.y  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
Assertion
Ref Expression
aomclem2  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
Distinct variable groups:    y, z,
a, b, c, d    ph, a
Allowed substitution hints:    ph( y, z, b, c, d)    A( y, z, a, b, c, d)    B( y, z, a, b, c, d)    C( y, z, a, b, c, d)

Proof of Theorem aomclem2
StepHypRef Expression
1 vex 2793 . . . . 5  |-  a  e. 
_V
2 aomclem2.y . . . . . . . . . 10  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
3 aomclem2.on . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  z  e.  On )
4 aomclem2.a . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  On )
53, 4jca 518 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dom  z  e.  On  /\  A  e.  On ) )
6 aomclem2.za . . . . . . . . . . . . 13  |-  ( ph  ->  dom  z  C_  A
)
7 r1ord3 7456 . . . . . . . . . . . . 13  |-  ( ( dom  z  e.  On  /\  A  e.  On )  ->  ( dom  z  C_  A  ->  ( R1 ` 
dom  z )  C_  ( R1 `  A ) ) )
85, 6, 7sylc 56 . . . . . . . . . . . 12  |-  ( ph  ->  ( R1 `  dom  z )  C_  ( R1 `  A ) )
9 sspwb 4225 . . . . . . . . . . . 12  |-  ( ( R1 `  dom  z
)  C_  ( R1 `  A )  <->  ~P ( R1 `  dom  z ) 
C_  ~P ( R1 `  A ) )
108, 9sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  ~P ( R1 `  dom  z )  C_  ~P ( R1 `  A ) )
1110sseld 3181 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  a  e.  ~P ( R1 `  A
) ) )
12 rsp 2605 . . . . . . . . . 10  |-  ( A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) )  -> 
( a  e.  ~P ( R1 `  A )  ->  ( a  =/=  (/)  ->  ( y `  a )  e.  ( ( ~P a  i^i 
Fin )  \  { (/)
} ) ) ) )
132, 11, 12sylsyld 52 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  ( a  =/=  (/)  ->  ( y `  a )  e.  ( ( ~P a  i^i 
Fin )  \  { (/)
} ) ) ) )
14133imp 1145 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) )
15 eldifi 3300 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/) } )  -> 
( y `  a
)  e.  ( ~P a  i^i  Fin )
)
1614, 15syl 15 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ~P a  i^i 
Fin ) )
17 inss1 3391 . . . . . . . . 9  |-  ( ~P a  i^i  Fin )  C_ 
~P a
1817sseli 3178 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  e.  ~P a )
19 elpwi 3635 . . . . . . . 8  |-  ( ( y `  a )  e.  ~P a  -> 
( y `  a
)  C_  a )
2018, 19syl 15 . . . . . . 7  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  C_  a )
2116, 20syl 15 . . . . . 6  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  a )
22 aomclem2.b . . . . . . . . 9  |-  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 ) ) ) }
23 aomclem2.su . . . . . . . . 9  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
24 aomclem2.we . . . . . . . . 9  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
2522, 3, 23, 24aomclem1 27162 . . . . . . . 8  |-  ( ph  ->  B  Or  ( R1
`  dom  z )
)
26253ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  B  Or  ( R1 `  dom  z ) )
27 inss2 3392 . . . . . . . 8  |-  ( ~P a  i^i  Fin )  C_ 
Fin
2827, 16sseldi 3180 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  Fin )
29 eldifsni 3752 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/) } )  -> 
( y `  a
)  =/=  (/) )
3014, 29syl 15 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  =/=  (/) )
31 elpwi 3635 . . . . . . . . 9  |-  ( a  e.  ~P ( R1
`  dom  z )  ->  a  C_  ( R1 ` 
dom  z ) )
32313ad2ant2 977 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  a  C_  ( R1 `  dom  z ) )
3321, 32sstrd 3191 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  ( R1 `  dom  z ) )
34 fisupcl 7220 . . . . . . 7  |-  ( ( B  Or  ( R1
`  dom  z )  /\  ( ( y `  a )  e.  Fin  /\  ( y `  a
)  =/=  (/)  /\  (
y `  a )  C_  ( R1 `  dom  z ) ) )  ->  sup ( ( y `
 a ) ,  ( R1 `  dom  z ) ,  B
)  e.  ( y `
 a ) )
3526, 28, 30, 33, 34syl13anc 1184 . . . . . 6  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  ( y `  a ) )
3621, 35sseldd 3183 . . . . 5  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  a )
37 aomclem2.c . . . . . 6  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
3837fvmpt2 5610 . . . . 5  |-  ( ( a  e.  _V  /\  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  a )  -> 
( C `  a
)  =  sup (
( y `  a
) ,  ( R1
`  dom  z ) ,  B ) )
391, 36, 38sylancr 644 . . . 4  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  ( C `  a )  =  sup ( ( y `
 a ) ,  ( R1 `  dom  z ) ,  B
) )
4039, 36eqeltrd 2359 . . 3  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  ( C `  a )  e.  a )
41403exp 1150 . 2  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  ( a  =/=  (/)  ->  ( C `  a )  e.  a ) ) )
4241ralrimiv 2627 1  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790    \ cdif 3151    i^i cin 3153    C_ wss 3154   (/)c0 3457   ~Pcpw 3627   {csn 3642   U.cuni 3829   class class class wbr 4025   {copab 4078    e. cmpt 4079    Or wor 4315    We wwe 4353   Oncon0 4394   suc csuc 4396   dom cdm 4691   ` cfv 5257   Fincfn 6865   supcsup 7195   R1cr1 7436
This theorem is referenced by:  aomclem3  27164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-er 6662  df-map 6776  df-en 6866  df-fin 6869  df-sup 7196  df-r1 7438
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