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Theorem uniimadomf 8959
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8958 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
Hypotheses
Ref Expression
uniimadomf.1  |-  F/_ x F
uniimadomf.2  |-  A  e. 
_V
uniimadomf.3  |-  B  e. 
_V
Assertion
Ref Expression
uniimadomf  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    F( x)

Proof of Theorem uniimadomf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1751 . . 3  |-  F/ z ( F `  x
)  ~<_  B
2 uniimadomf.1 . . . . 5  |-  F/_ x F
3 nfcv 2582 . . . . 5  |-  F/_ x
z
42, 3nffv 5879 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2582 . . . 4  |-  F/_ x  ~<_
6 nfcv 2582 . . . 4  |-  F/_ x B
74, 5, 6nfbr 4461 . . 3  |-  F/ x
( F `  z
)  ~<_  B
8 fveq2 5872 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
98breq1d 4427 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  ~<_  B  <->  ( F `  z )  ~<_  B ) )
101, 7, 9cbvral 3049 . 2  |-  ( A. x  e.  A  ( F `  x )  ~<_  B 
<-> 
A. z  e.  A  ( F `  z )  ~<_  B )
11 uniimadomf.2 . . 3  |-  A  e. 
_V
12 uniimadomf.3 . . 3  |-  B  e. 
_V
1311, 12uniimadom 8958 . 2  |-  ( ( Fun  F  /\  A. z  e.  A  ( F `  z )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
1410, 13sylan2b 477 1  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1867   F/_wnfc 2568   A.wral 2773   _Vcvv 3078   U.cuni 4213   class class class wbr 4417    X. cxp 4843   "cima 4848   Fun wfun 5586   ` cfv 5592    ~<_ cdom 7566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-ac2 8882
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-card 8363  df-acn 8366  df-ac 8536
This theorem is referenced by: (None)
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