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Theorem uniimadomf 8163
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8162 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 2420 . . . . 5  |-  F/_ x
z
42, 3nffv 5493 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2420 . . . 4  |-  F/_ x  ~<_
6 nfcv 2420 . . . 4  |-  F/_ x B
74, 5, 6nfbr 4068 . . 3  |-  F/ x
( F `  z
)  ~<_  B
8 fveq2 5486 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
98breq1d 4034 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  ~<_  B  <->  ( F `  z )  ~<_  B ) )
101, 7, 9cbvral 2761 . 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 8162 . 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 1685   F/_wnfc 2407   A.wral 2544   _Vcvv 2789   U.cuni 3828   class class class wbr 4024    X. cxp 4686   "cima 4691   Fun wfun 5215   ` cfv 5221    ~<_ cdom 6857
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8085
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-card 7568  df-acn 7571  df-ac 7739
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