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Theorem unirnfdomd 8369
Description: The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
unirnfdomd.1  |-  ( ph  ->  F : T --> Fin )
unirnfdomd.2  |-  ( ph  ->  -.  T  e.  Fin )
unirnfdomd.3  |-  ( ph  ->  T  e.  V )
Assertion
Ref Expression
unirnfdomd  |-  ( ph  ->  U. ran  F  ~<_  T )

Proof of Theorem unirnfdomd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 unirnfdomd.1 . . . . . . . 8  |-  ( ph  ->  F : T --> Fin )
2 ffn 5525 . . . . . . . 8  |-  ( F : T --> Fin  ->  F  Fn  T )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  T )
4 unirnfdomd.3 . . . . . . 7  |-  ( ph  ->  T  e.  V )
5 fnex 5894 . . . . . . 7  |-  ( ( F  Fn  T  /\  T  e.  V )  ->  F  e.  _V )
63, 4, 5syl2anc 643 . . . . . 6  |-  ( ph  ->  F  e.  _V )
7 rnexg 5065 . . . . . 6  |-  ( F  e.  _V  ->  ran  F  e.  _V )
86, 7syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  _V )
9 frn 5531 . . . . . . 7  |-  ( F : T --> Fin  ->  ran 
F  C_  Fin )
10 dfss3 3275 . . . . . . 7  |-  ( ran 
F  C_  Fin  <->  A. x  e.  ran  F  x  e. 
Fin )
119, 10sylib 189 . . . . . 6  |-  ( F : T --> Fin  ->  A. x  e.  ran  F  x  e.  Fin )
12 isfinite 7534 . . . . . . . 8  |-  ( x  e.  Fin  <->  x  ~<  om )
13 sdomdom 7065 . . . . . . . 8  |-  ( x 
~<  om  ->  x  ~<_  om )
1412, 13sylbi 188 . . . . . . 7  |-  ( x  e.  Fin  ->  x  ~<_  om )
1514ralimi 2718 . . . . . 6  |-  ( A. x  e.  ran  F  x  e.  Fin  ->  A. x  e.  ran  F  x  ~<_  om )
161, 11, 153syl 19 . . . . 5  |-  ( ph  ->  A. x  e.  ran  F  x  ~<_  om )
17 unidom 8345 . . . . 5  |-  ( ( ran  F  e.  _V  /\ 
A. x  e.  ran  F  x  ~<_  om )  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
188, 16, 17syl2anc 643 . . . 4  |-  ( ph  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
19 fnrndomg 8340 . . . . . 6  |-  ( T  e.  V  ->  ( F  Fn  T  ->  ran 
F  ~<_  T ) )
204, 3, 19sylc 58 . . . . 5  |-  ( ph  ->  ran  F  ~<_  T )
21 omex 7525 . . . . . 6  |-  om  e.  _V
2221xpdom1 7137 . . . . 5  |-  ( ran 
F  ~<_  T  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om )
)
2320, 22syl 16 . . . 4  |-  ( ph  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )
24 domtr 7090 . . . 4  |-  ( ( U. ran  F  ~<_  ( ran  F  X.  om )  /\  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )  ->  U. ran  F  ~<_  ( T  X.  om )
)
2518, 23, 24syl2anc 643 . . 3  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  om )
)
26 unirnfdomd.2 . . . . 5  |-  ( ph  ->  -.  T  e.  Fin )
27 infinf 8368 . . . . . 6  |-  ( T  e.  V  ->  ( -.  T  e.  Fin  <->  om  ~<_  T ) )
284, 27syl 16 . . . . 5  |-  ( ph  ->  ( -.  T  e. 
Fin 
<->  om  ~<_  T ) )
2926, 28mpbid 202 . . . 4  |-  ( ph  ->  om  ~<_  T )
30 xpdom2g 7134 . . . 4  |-  ( ( T  e.  V  /\  om  ~<_  T )  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
314, 29, 30syl2anc 643 . . 3  |-  ( ph  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
32 domtr 7090 . . 3  |-  ( ( U. ran  F  ~<_  ( T  X.  om )  /\  ( T  X.  om )  ~<_  ( T  X.  T ) )  ->  U. ran  F  ~<_  ( T  X.  T ) )
3325, 31, 32syl2anc 643 . 2  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  T ) )
34 infxpidm 8364 . . 3  |-  ( om  ~<_  T  ->  ( T  X.  T )  ~~  T
)
3529, 34syl 16 . 2  |-  ( ph  ->  ( T  X.  T
)  ~~  T )
36 domentr 7096 . 2  |-  ( ( U. ran  F  ~<_  ( T  X.  T )  /\  ( T  X.  T )  ~~  T
)  ->  U. ran  F  ~<_  T )
3733, 35, 36syl2anc 643 1  |-  ( ph  ->  U. ran  F  ~<_  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    e. wcel 1717   A.wral 2643   _Vcvv 2893    C_ wss 3257   U.cuni 3951   class class class wbr 4147   omcom 4779    X. cxp 4810   ran crn 4813    Fn wfn 5383   -->wf 5384    ~~ cen 7036    ~<_ cdom 7037    ~< csdm 7038   Fincfn 7039
This theorem is referenced by:  acsdomd  14528
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-inf2 7523  ax-ac2 8270
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-se 4477  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-isom 5397  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-map 6950  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-oi 7406  df-card 7753  df-acn 7756  df-ac 7924
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