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Theorem iundomg 8417
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 8416 . . . 4  |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
5 iundomg.4 . . . 4  |-  ( ph  ->  ( A  X.  C
)  e. AC  U_ x  e.  A  B )
6 acndom2 7936 . . . 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 59 . . 3  |-  ( ph  ->  T  e. AC  U_ x  e.  A  B )
81iunfo 8415 . . 3  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
9 fodomacn 7938 . . 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 1386 . 2  |-  ( ph  ->  U_ x  e.  A  B  ~<_  T )
11 domtr 7161 . 2  |-  ( (
U_ x  e.  A  B  ~<_  T  /\  T  ~<_  ( A  X.  C
) )  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
1210, 4, 11syl2anc 644 1  |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2706   {csn 3815   U_ciun 4094   class class class wbr 4213    X. cxp 4877    |` cres 4881   -onto->wfo 5453  (class class class)co 6082   2ndc2nd 6349    ^m cmap 7019    ~<_ cdom 7108  AC wacn 7826
This theorem is referenced by:  iundom  8418  iunctb  8450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-map 7021  df-dom 7112  df-acn 7830
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