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Theorem aomclem2 27121
Description: Lemma for dfac11 27128. 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 2951 . . . . 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 519 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dom  z  e.  On  /\  A  e.  On ) )
6 aomclem2.za . . . . . . . . . . . . 13  |-  ( ph  ->  dom  z  C_  A
)
7 r1ord3 7700 . . . . . . . . . . . . 13  |-  ( ( dom  z  e.  On  /\  A  e.  On )  ->  ( dom  z  C_  A  ->  ( R1 ` 
dom  z )  C_  ( R1 `  A ) ) )
85, 6, 7sylc 58 . . . . . . . . . . . 12  |-  ( ph  ->  ( R1 `  dom  z )  C_  ( R1 `  A ) )
9 sspwb 4405 . . . . . . . . . . . 12  |-  ( ( R1 `  dom  z
)  C_  ( R1 `  A )  <->  ~P ( R1 `  dom  z ) 
C_  ~P ( R1 `  A ) )
108, 9sylib 189 . . . . . . . . . . 11  |-  ( ph  ->  ~P ( R1 `  dom  z )  C_  ~P ( R1 `  A ) )
1110sseld 3339 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  a  e.  ~P ( R1 `  A
) ) )
12 rsp 2758 . . . . . . . . . 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 54 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  ( a  =/=  (/)  ->  ( y `  a )  e.  ( ( ~P a  i^i 
Fin )  \  { (/)
} ) ) ) )
14133imp 1147 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) )
1514eldifad 3324 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ~P a  i^i 
Fin ) )
16 inss1 3553 . . . . . . . . 9  |-  ( ~P a  i^i  Fin )  C_ 
~P a
1716sseli 3336 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  e.  ~P a )
1817elpwid 3800 . . . . . . 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 27120 . . . . . . . 8  |-  ( ph  ->  B  Or  ( R1
`  dom  z )
)
24233ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  B  Or  ( R1 `  dom  z ) )
25 inss2 3554 . . . . . . . 8  |-  ( ~P a  i^i  Fin )  C_ 
Fin
2625, 15sseldi 3338 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  Fin )
27 eldifsni 3920 . . . . . . . 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 3799 . . . . . . . . 9  |-  ( a  e.  ~P ( R1
`  dom  z )  ->  a  C_  ( R1 ` 
dom  z ) )
30293ad2ant2 979 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  a  C_  ( R1 `  dom  z ) )
3119, 30sstrd 3350 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  ( R1 `  dom  z ) )
32 fisupcl 7464 . . . . . . 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 1186 . . . . . 6  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B )  e.  ( y `  a ) )
3419, 33sseldd 3341 . . . . 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 5804 . . . . 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 645 . . . 4  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  ( C `  a )  =  sup ( ( y `
 a ) ,  ( R1 `  dom  z ) ,  B
) )
3837, 34eqeltrd 2509 . . 3  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  ( C `  a )  e.  a )
39383exp 1152 . 2  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  ( a  =/=  (/)  ->  ( C `  a )  e.  a ) ) )
4039ralrimiv 2780 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 177    /\ wa 359    /\ w3a 936    = 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   class class class wbr 4204   {copab 4257    e. cmpt 4258    Or wor 4494    We wwe 4532   Oncon0 4573   suc csuc 4575   dom cdm 4870   ` cfv 5446   Fincfn 7101   supcsup 7437   R1cr1 7680
This theorem is referenced by:  aomclem3  27122
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-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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