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Theorem iundomg 8820
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 8819 . . . 4  |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
5 iundomg.4 . . . 4  |-  ( ph  ->  ( A  X.  C
)  e. AC  U_ x  e.  A  B )
6 acndom2 8339 . . . 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 60 . . 3  |-  ( ph  ->  T  e. AC  U_ x  e.  A  B )
81iunfo 8818 . . 3  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
9 fodomacn 8341 . . 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, 9mpisyl 18 . 2  |-  ( ph  ->  U_ x  e.  A  B  ~<_  T )
11 domtr 7475 . 2  |-  ( (
U_ x  e.  A  B  ~<_  T  /\  T  ~<_  ( A  X.  C
) )  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
1210, 4, 11syl2anc 661 1  |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2799   {csn 3988   U_ciun 4282   class class class wbr 4403    X. cxp 4949    |` cres 4953   -onto->wfo 5527  (class class class)co 6203   2ndc2nd 6689    ^m cmap 7327    ~<_ cdom 7421  AC wacn 8223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-dom 7425  df-acn 8227
This theorem is referenced by:  iundom  8821  iunctb  8853
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