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Theorem uniimadomf 8167
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8166 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 1605 . . 3  |-  F/ z ( F `  x
)  ~<_  B
2 uniimadomf.1 . . . . 5  |-  F/_ x F
3 nfcv 2419 . . . . 5  |-  F/_ x
z
42, 3nffv 5532 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2419 . . . 4  |-  F/_ x  ~<_
6 nfcv 2419 . . . 4  |-  F/_ x B
74, 5, 6nfbr 4067 . . 3  |-  F/ x
( F `  z
)  ~<_  B
8 fveq2 5525 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
98breq1d 4033 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  ~<_  B  <->  ( F `  z )  ~<_  B ) )
101, 7, 9cbvral 2760 . 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 8166 . 2  |-  ( ( Fun  F  /\  A. z  e.  A  ( F `  z )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
1410, 13sylan2b 461 1  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   F/_wnfc 2406   A.wral 2543   _Vcvv 2788   U.cuni 3827   class class class wbr 4023    X. cxp 4687   "cima 4692   Fun wfun 5249   ` cfv 5255    ~<_ cdom 6861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-card 7572  df-acn 7575  df-ac 7743
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