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Theorem iundomg 8163
Description: An upper bound for the cardinality of an indexed union, with explicit choice principles.  B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
iunfo.1  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
iundomg.2  |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )
iundomg.3  |-  ( ph  ->  A. x  e.  A  B  ~<_  C )
iundomg.4  |-  ( ph  ->  ( A  X.  C
)  e. AC  U_ x  e.  A  B )
Assertion
Ref Expression
iundomg  |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    ph( x)    B( x)    T( x)

Proof of Theorem iundomg
StepHypRef Expression
1 iunfo.1 . . . . 5  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
2 iundomg.2 . . . . 5  |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )
3 iundomg.3 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  ~<_  C )
41, 2, 3iundom2g 8162 . . . 4  |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
5 iundomg.4 . . . 4  |-  ( ph  ->  ( A  X.  C
)  e. AC  U_ x  e.  A  B )
6 acndom2 7681 . . . 4  |-  ( T  ~<_  ( A  X.  C
)  ->  ( ( A  X.  C )  e. AC  U_ x  e.  A  B  ->  T  e. AC  U_ x  e.  A  B ) )
74, 5, 6sylc 56 . . 3  |-  ( ph  ->  T  e. AC  U_ x  e.  A  B )
81iunfo 8161 . . 3  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
9 fodomacn 7683 . . 3  |-  ( T  e. AC  U_ x  e.  A  B  ->  ( ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B  ->  U_ x  e.  A  B  ~<_  T ) )
107, 8, 9ee10 1366 . 2  |-  ( ph  ->  U_ x  e.  A  B  ~<_  T )
11 domtr 6914 . 2  |-  ( (
U_ x  e.  A  B  ~<_  T  /\  T  ~<_  ( A  X.  C
) )  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
1210, 4, 11syl2anc 642 1  |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640   U_ciun 3905   class class class wbr 4023    X. cxp 4687    |` cres 4691   -onto->wfo 5253  (class class class)co 5858   2ndc2nd 6121    ^m cmap 6772    ~<_ cdom 6861  AC wacn 7571
This theorem is referenced by:  iundom  8164  iunctb  8196
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-dom 6865  df-acn 7575
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