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Theorem uniimadomf 8380
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8379 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 1626 . . 3  |-  F/ z ( F `  x
)  ~<_  B
2 uniimadomf.1 . . . . 5  |-  F/_ x F
3 nfcv 2544 . . . . 5  |-  F/_ x
z
42, 3nffv 5698 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2544 . . . 4  |-  F/_ x  ~<_
6 nfcv 2544 . . . 4  |-  F/_ x B
74, 5, 6nfbr 4220 . . 3  |-  F/ x
( F `  z
)  ~<_  B
8 fveq2 5691 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
98breq1d 4186 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  ~<_  B  <->  ( F `  z )  ~<_  B ) )
101, 7, 9cbvral 2892 . 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 8379 . 2  |-  ( ( Fun  F  /\  A. z  e.  A  ( F `  z )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
1410, 13sylan2b 462 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 359    e. wcel 1721   F/_wnfc 2531   A.wral 2670   _Vcvv 2920   U.cuni 3979   class class class wbr 4176    X. cxp 4839   "cima 4844   Fun wfun 5411   ` cfv 5417    ~<_ cdom 7070
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-ac2 8303
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-card 7786  df-acn 7789  df-ac 7957
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