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Theorem iundomg 8947
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 8946 . . . 4  |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
5 iundomg.4 . . . 4  |-  ( ph  ->  ( A  X.  C
)  e. AC  U_ x  e.  A  B )
6 acndom2 8466 . . . 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 8945 . . 3  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
9 fodomacn 8468 . . 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 19 . 2  |-  ( ph  ->  U_ x  e.  A  B  ~<_  T )
11 domtr 7605 . 2  |-  ( (
U_ x  e.  A  B  ~<_  T  /\  T  ~<_  ( A  X.  C
) )  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
1210, 4, 11syl2anc 659 1  |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   A.wral 2753   {csn 3971   U_ciun 4270   class class class wbr 4394    X. cxp 4820    |` cres 4824   -onto->wfo 5566  (class class class)co 6277   2ndc2nd 6782    ^m cmap 7456    ~<_ cdom 7551  AC wacn 8350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-map 7458  df-dom 7555  df-acn 8354
This theorem is referenced by:  iundom  8948  iunctb  8980
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