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Theorem marypha2 7935
Description: Version of marypha1 7930 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 7848 . . 3  |-  ( ph  ->  U. ran  F  e. 
Fin )
4 eqid 2404 . . . . 5  |-  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  =  U_ x  e.  A  ( {
x }  X.  ( F `  x )
)
54marypha2lem1 7931 . . . 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 5716 . . . . . 6  |-  ( F : A --> Fin  ->  F  Fn  A )
92, 8syl 17 . . . . 5  |-  ( ph  ->  F  Fn  A )
104marypha2lem4 7934 . . . . 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 4423 . . 3  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  ( U_ x  e.  A  ( { x }  X.  ( F `  x ) ) " d ) )
131, 3, 6, 12marypha1 7930 . 2  |-  ( ph  ->  E. g  e.  ~P  U_ x  e.  A  ( { x }  X.  ( F `  x ) ) g : A -1-1-> U.
ran  F )
14 df-rex 2762 . . 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 3464 . . . . . . . 8  |-  U. ran  F 
C_  _V
16 f1ss 5771 . . . . . . . 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 729 . . . . . 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 3966 . . . . . . . 8  |-  ( g  e.  ~P U_ x  e.  A  ( {
x }  X.  ( F `  x )
)  ->  g  C_  U_ x  e.  A  ( { x }  X.  ( F `  x ) ) )
2019ad2antrl 728 . . . . . . 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 5767 . . . . . . . . 9  |-  ( g : A -1-1-> U. ran  F  ->  g  Fn  A
)
2322ad2antll 729 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ~P U_ x  e.  A  ( { x }  X.  ( F `  x ) )  /\  g : A -1-1-> U. ran  F ) )  ->  g  Fn  A )
244marypha2lem3 7933 . . . . . . . 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 212 . . . . . 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 1733 . . 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 219 . 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 186    /\ wa 369    = wceq 1407   E.wex 1635    e. wcel 1844   A.wral 2756   E.wrex 2757   _Vcvv 3061    C_ wss 3416   ~Pcpw 3957   {csn 3974   U.cuni 4193   U_ciun 4273   class class class wbr 4397    X. cxp 4823   ran crn 4826   "cima 4828    Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   ` cfv 5571    ~<_ cdom 7554   Fincfn 7556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560
This theorem is referenced by: (None)
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