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Theorem uniimadomf 8169
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8168 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 1607 . . 3  |-  F/ z ( F `  x
)  ~<_  B
2 uniimadomf.1 . . . . 5  |-  F/_ x F
3 nfcv 2421 . . . . 5  |-  F/_ x
z
42, 3nffv 5534 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2421 . . . 4  |-  F/_ x  ~<_
6 nfcv 2421 . . . 4  |-  F/_ x B
74, 5, 6nfbr 4069 . . 3  |-  F/ x
( F `  z
)  ~<_  B
8 fveq2 5527 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
98breq1d 4035 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  ~<_  B  <->  ( F `  z )  ~<_  B ) )
101, 7, 9cbvral 2762 . 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 8168 . 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 1625    e. wcel 1686   F/_wnfc 2408   A.wral 2545   _Vcvv 2790   U.cuni 3829   class class class wbr 4025    X. cxp 4689   "cima 4694   Fun wfun 5251   ` cfv 5257    ~<_ cdom 6863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-ac2 8091
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-card 7574  df-acn 7577  df-ac 7745
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