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Theorem marypha2 7899
Description: Version of marypha1 7894 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypotheses
Ref Expression
marypha2.a  |-  ( ph  ->  A  e.  Fin )
marypha2.b  |-  ( ph  ->  F : A --> Fin )
marypha2.c  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  U. ( F " d
) )
Assertion
Ref Expression
marypha2  |-  ( ph  ->  E. g ( g : A -1-1-> _V  /\  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
Distinct variable groups:    ph, d, g, x    A, d, g, x    F, d, g, x

Proof of Theorem marypha2
StepHypRef Expression
1 marypha2.a . . 3  |-  ( ph  ->  A  e.  Fin )
2 marypha2.b . . . 4  |-  ( ph  ->  F : A --> Fin )
32, 1unirnffid 7812 . . 3  |-  ( ph  ->  U. ran  F  e. 
Fin )
4 eqid 2467 . . . . 5  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)
54marypha2lem1 7895 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  C_  ( A  X.  U. ran  F )
65a1i 11 . . 3  |-  ( ph  ->  U_ x  e.  A  ( { x }  X.  ( F `  x ) )  C_  ( A  X.  U. ran  F ) )
7 marypha2.c . . . 4  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  U. ( F " d
) )
8 ffn 5731 . . . . . 6  |-  ( F : A --> Fin  ->  F  Fn  A )
92, 8syl 16 . . . . 5  |-  ( ph  ->  F  Fn  A )
104marypha2lem4 7898 . . . . 5  |-  ( ( F  Fn  A  /\  d  C_  A )  -> 
( U_ x  e.  A  ( { x }  X.  ( F `  x ) ) " d )  =  U. ( F
" d ) )
119, 10sylan 471 . . . 4  |-  ( (
ph  /\  d  C_  A )  ->  ( U_ x  e.  A  ( { x }  X.  ( F `  x ) ) " d )  =  U. ( F
" d ) )
127, 11breqtrrd 4473 . . 3  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  ( U_ x  e.  A  ( { x }  X.  ( F `  x ) ) " d ) )
131, 3, 6, 12marypha1 7894 . 2  |-  ( ph  ->  E. g  e.  ~P  U_ x  e.  A  ( { x }  X.  ( F `  x ) ) g : A -1-1-> U.
ran  F )
14 df-rex 2820 . . 3  |-  ( E. g  e.  ~P  U_ x  e.  A  ( { x }  X.  ( F `  x ) ) g : A -1-1-> U.
ran  F  <->  E. g ( g  e.  ~P U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  /\  g : A -1-1-> U. ran  F ) )
15 ssv 3524 . . . . . . . 8  |-  U. ran  F 
C_  _V
16 f1ss 5786 . . . . . . . 8  |-  ( ( g : A -1-1-> U. ran  F  /\  U. ran  F 
C_  _V )  ->  g : A -1-1-> _V )
1715, 16mpan2 671 . . . . . . 7  |-  ( g : A -1-1-> U. ran  F  ->  g : A -1-1-> _V )
1817ad2antll 728 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  g : A -1-1-> _V )
19 elpwi 4019 . . . . . . . 8  |-  ( g  e.  ~P U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  ->  g  C_  U_ x  e.  A  ( { x }  X.  ( F `  x ) ) )
2019ad2antrl 727 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  g  C_ 
U_ x  e.  A  ( { x }  X.  ( F `  x ) ) )
219adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  F  Fn  A )
22 f1fn 5782 . . . . . . . . 9  |-  ( g : A -1-1-> U. ran  F  ->  g  Fn  A
)
2322ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  g  Fn  A )
244marypha2lem3 7897 . . . . . . . 8  |-  ( ( F  Fn  A  /\  g  Fn  A )  ->  ( g  C_  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  <->  A. x  e.  A  ( g `  x
)  e.  ( F `
 x ) ) )
2521, 23, 24syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  (
g  C_  U_ x  e.  A  ( { x }  X.  ( F `  x ) )  <->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
2620, 25mpbid 210 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  A. x  e.  A  ( g `  x )  e.  ( F `  x ) )
2718, 26jca 532 . . . . 5  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  (
g : A -1-1-> _V  /\ 
A. x  e.  A  ( g `  x
)  e.  ( F `
 x ) ) )
2827ex 434 . . . 4  |-  ( ph  ->  ( ( g  e. 
~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F )  ->  ( g : A -1-1-> _V  /\  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) ) )
2928eximdv 1686 . . 3  |-  ( ph  ->  ( E. g ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F )  ->  E. g ( g : A -1-1-> _V  /\  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) ) )
3014, 29syl5bi 217 . 2  |-  ( ph  ->  ( E. g  e. 
~P  U_ x  e.  A  ( { x }  X.  ( F `  x ) ) g : A -1-1-> U.
ran  F  ->  E. g
( g : A -1-1-> _V 
/\  A. x  e.  A  ( g `  x
)  e.  ( F `
 x ) ) ) )
3113, 30mpd 15 1  |-  ( ph  ->  E. g ( g : A -1-1-> _V  /\  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   U_ciun 4325   class class class wbr 4447    X. cxp 4997   ran crn 5000   "cima 5002    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   ` cfv 5588    ~<_ cdom 7514   Fincfn 7516
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520
This theorem is referenced by: (None)
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