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Theorem aomclem2 30935
Description: Lemma for dfac11 30942. 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 3121 . . . . 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 532 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dom  z  e.  On  /\  A  e.  On ) )
6 aomclem2.za . . . . . . . . . . . . 13  |-  ( ph  ->  dom  z  C_  A
)
7 r1ord3 8212 . . . . . . . . . . . . 13  |-  ( ( dom  z  e.  On  /\  A  e.  On )  ->  ( dom  z  C_  A  ->  ( R1 ` 
dom  z )  C_  ( R1 `  A ) ) )
85, 6, 7sylc 60 . . . . . . . . . . . 12  |-  ( ph  ->  ( R1 `  dom  z )  C_  ( R1 `  A ) )
9 sspwb 4702 . . . . . . . . . . . 12  |-  ( ( R1 `  dom  z
)  C_  ( R1 `  A )  <->  ~P ( R1 `  dom  z ) 
C_  ~P ( R1 `  A ) )
108, 9sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  ~P ( R1 `  dom  z )  C_  ~P ( R1 `  A ) )
1110sseld 3508 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  a  e.  ~P ( R1 `  A
) ) )
12 rsp 2833 . . . . . . . . . 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 56 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  ( a  =/=  (/)  ->  ( y `  a )  e.  ( ( ~P a  i^i 
Fin )  \  { (/)
} ) ) ) )
14133imp 1190 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) )
1514eldifad 3493 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ~P a  i^i 
Fin ) )
16 inss1 3723 . . . . . . . . 9  |-  ( ~P a  i^i  Fin )  C_ 
~P a
1716sseli 3505 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  e.  ~P a )
1817elpwid 4026 . . . . . . 7  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  C_  a )
1915, 18syl 16 . . . . . 6  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  a )
20 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 ) ) ) }
21 aomclem2.su . . . . . . . . 9  |-  ( ph  ->  dom  z  =  suc  U.
dom  z )
22 aomclem2.we . . . . . . . . 9  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
2320, 3, 21, 22aomclem1 30934 . . . . . . . 8  |-  ( ph  ->  B  Or  ( R1
`  dom  z )
)
24233ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  B  Or  ( R1 `  dom  z ) )
25 inss2 3724 . . . . . . . 8  |-  ( ~P a  i^i  Fin )  C_ 
Fin
2625, 15sseldi 3507 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  Fin )
27 eldifsni 4159 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/) } )  -> 
( y `  a
)  =/=  (/) )
2814, 27syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  =/=  (/) )
29 elpwi 4025 . . . . . . . . 9  |-  ( a  e.  ~P ( R1
`  dom  z )  ->  a  C_  ( R1 ` 
dom  z ) )
30293ad2ant2 1018 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  a  C_  ( R1 `  dom  z ) )
3119, 30sstrd 3519 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  ( R1 `  dom  z ) )
32 fisupcl 7939 . . . . . . 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 ) )
3324, 26, 28, 31, 32syl13anc 1230 . . . . . 6  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  ( y `  a ) )
3419, 33sseldd 3510 . . . . 5  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  a )
35 aomclem2.c . . . . . 6  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
3635fvmpt2 5964 . . . . 5  |-  ( ( a  e.  _V  /\  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  a )  -> 
( C `  a
)  =  sup (
( y `  a
) ,  ( R1
`  dom  z ) ,  B ) )
371, 34, 36sylancr 663 . . . 4  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  ( C `  a )  =  sup ( ( y `
 a ) ,  ( R1 `  dom  z ) ,  B
) )
3837, 34eqeltrd 2555 . . 3  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  ( C `  a )  e.  a )
39383exp 1195 . 2  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  ( a  =/=  (/)  ->  ( C `  a )  e.  a ) ) )
4039ralrimiv 2879 1  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  dom  z
) ( a  =/=  (/)  ->  ( C `  a )  e.  a ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   U.cuni 4251   class class class wbr 4453   {copab 4510    |-> cmpt 4511    Or wor 4805    We wwe 4843   Oncon0 4884   suc csuc 4886   dom cdm 5005   ` cfv 5594   Fincfn 7528   supcsup 7912   R1cr1 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-er 7323  df-map 7434  df-en 7529  df-fin 7532  df-sup 7913  df-r1 8194
This theorem is referenced by:  aomclem3  30936
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