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Theorem iundomg 8179
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 8178 . . . 4  |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
5 iundomg.4 . . . 4  |-  ( ph  ->  ( A  X.  C
)  e. AC  U_ x  e.  A  B )
6 acndom2 7697 . . . 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 8177 . . 3  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
9 fodomacn 7699 . . 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 6930 . 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 1632    e. wcel 1696   A.wral 2556   {csn 3653   U_ciun 3921   class class class wbr 4039    X. cxp 4703    |` cres 4707   -onto->wfo 5269  (class class class)co 5874   2ndc2nd 6137    ^m cmap 6788    ~<_ cdom 6877  AC wacn 7587
This theorem is referenced by:  iundom  8180  iunctb  8212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-dom 6881  df-acn 7591
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