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Theorem aomclem2 26475
Description: Lemma for dfac11 26483. 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 2867 . . . . 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 7541 . . . . . . . . . . . . 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 4302 . . . . . . . . . . . 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 3255 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  a  e.  ~P ( R1 `  A
) ) )
12 rsp 2679 . . . . . . . . . 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 3374 . . . . . . . 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 3465 . . . . . . . . 9  |-  ( ~P a  i^i  Fin )  C_ 
~P a
1817sseli 3252 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  e.  ~P a )
19 elpwi 3709 . . . . . . . 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 26474 . . . . . . . 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 3466 . . . . . . . 8  |-  ( ~P a  i^i  Fin )  C_ 
Fin
2827, 16sseldi 3254 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  Fin )
29 eldifsni 3826 . . . . . . . 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 3709 . . . . . . . . 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 3265 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  ( R1 `  dom  z ) )
34 fisupcl 7305 . . . . . . 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 3257 . . . . 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 5688 . . . . 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 2432 . . 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 2701 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 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   _Vcvv 2864    \ cdif 3225    i^i cin 3227    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   {csn 3716   U.cuni 3906   class class class wbr 4102   {copab 4155    e. cmpt 4156    Or wor 4392    We wwe 4430   Oncon0 4471   suc csuc 4473   dom cdm 4768   ` cfv 5334   Fincfn 6948   supcsup 7280   R1cr1 7521
This theorem is referenced by:  aomclem3  26476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-er 6744  df-map 6859  df-en 6949  df-fin 6952  df-sup 7281  df-r1 7523
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