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Theorem aomclem2 27028
Description: Lemma for dfac11 27036. 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 2927 . . . . 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 7672 . . . . . . . . . . . . 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 4381 . . . . . . . . . . . 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 3315 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  ~P ( R1 `  dom  z
)  ->  a  e.  ~P ( R1 `  A
) ) )
12 rsp 2734 . . . . . . . . . 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 3300 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  ( ~P a  i^i 
Fin ) )
16 inss1 3529 . . . . . . . . 9  |-  ( ~P a  i^i  Fin )  C_ 
~P a
1716sseli 3312 . . . . . . . 8  |-  ( ( y `  a )  e.  ( ~P a  i^i  Fin )  ->  (
y `  a )  e.  ~P a )
1817elpwid 3776 . . . . . . 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 27027 . . . . . . . 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 3530 . . . . . . . 8  |-  ( ~P a  i^i  Fin )  C_ 
Fin
2625, 15sseldi 3314 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  e.  Fin )
27 eldifsni 3896 . . . . . . . 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 3775 . . . . . . . . 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 3326 . . . . . . 7  |-  ( (
ph  /\  a  e.  ~P ( R1 `  dom  z )  /\  a  =/=  (/) )  ->  (
y `  a )  C_  ( R1 `  dom  z ) )
32 fisupcl 7436 . . . . . . 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 3317 . . . . 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 5779 . . . . 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 2486 . . 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 2756 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 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   _Vcvv 2924    \ cdif 3285    i^i cin 3287    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   {csn 3782   U.cuni 3983   class class class wbr 4180   {copab 4233    e. cmpt 4234    Or wor 4470    We wwe 4508   Oncon0 4549   suc csuc 4551   dom cdm 4845   ` cfv 5421   Fincfn 7076   supcsup 7411   R1cr1 7652
This theorem is referenced by:  aomclem3  27029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-er 6872  df-map 6987  df-en 7077  df-fin 7080  df-sup 7412  df-r1 7654
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