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Theorem iunfo 8177
Description: Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
Hypothesis
Ref Expression
iunfo.1  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
Assertion
Ref Expression
iunfo  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    T( x)

Proof of Theorem iunfo
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fo2nd 6156 . . . 4  |-  2nd : _V -onto-> _V
2 fof 5467 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
3 ffn 5405 . . . 4  |-  ( 2nd
: _V --> _V  ->  2nd 
Fn  _V )
41, 2, 3mp2b 9 . . 3  |-  2nd  Fn  _V
5 ssv 3211 . . 3  |-  T  C_  _V
6 fnssres 5373 . . 3  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( 2nd  |`  T )  Fn  T )
74, 5, 6mp2an 653 . 2  |-  ( 2nd  |`  T )  Fn  T
8 df-ima 4718 . . 3  |-  ( 2nd " T )  =  ran  ( 2nd  |`  T )
9 iunfo.1 . . . . . . . . . . 11  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
109eleq2i 2360 . . . . . . . . . 10  |-  ( z  e.  T  <->  z  e.  U_ x  e.  A  ( { x }  X.  B ) )
11 eliun 3925 . . . . . . . . . 10  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
1210, 11bitri 240 . . . . . . . . 9  |-  ( z  e.  T  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
13 xp2nd 6166 . . . . . . . . . . 11  |-  ( z  e.  ( { x }  X.  B )  -> 
( 2nd `  z
)  e.  B )
14 eleq1 2356 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  y  ->  (
( 2nd `  z
)  e.  B  <->  y  e.  B ) )
1513, 14syl5ib 210 . . . . . . . . . 10  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  ( { x }  X.  B
)  ->  y  e.  B ) )
1615reximdv 2667 . . . . . . . . 9  |-  ( ( 2nd `  z )  =  y  ->  ( E. x  e.  A  z  e.  ( {
x }  X.  B
)  ->  E. x  e.  A  y  e.  B ) )
1712, 16syl5bi 208 . . . . . . . 8  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  T  ->  E. x  e.  A  y  e.  B )
)
1817impcom 419 . . . . . . 7  |-  ( ( z  e.  T  /\  ( 2nd `  z )  =  y )  ->  E. x  e.  A  y  e.  B )
1918rexlimiva 2675 . . . . . 6  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  ->  E. x  e.  A  y  e.  B )
20 nfiu1 3949 . . . . . . . . 9  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
219, 20nfcxfr 2429 . . . . . . . 8  |-  F/_ x T
22 nfv 1609 . . . . . . . 8  |-  F/ x
( 2nd `  z
)  =  y
2321, 22nfrex 2611 . . . . . . 7  |-  F/ x E. z  e.  T  ( 2nd `  z )  =  y
24 ssiun2 3961 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
2524adantr 451 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
26 simpr 447 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  B )
27 vex 2804 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2827snid 3680 . . . . . . . . . . . . 13  |-  x  e. 
{ x }
29 opelxp 4735 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  ( x  e.  { x }  /\  y  e.  B )
)
3028, 29mpbiran 884 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  y  e.  B )
3126, 30sylibr 203 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( { x }  X.  B ) )
3225, 31sseldd 3194 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B ) )
3332, 9syl6eleqr 2387 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  T )
34 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
3527, 34op2nd 6145 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
36 fveq2 5541 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  ( 2nd `  <. x ,  y
>. ) )
3736eqeq1d 2304 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  z )  =  y  <-> 
( 2nd `  <. x ,  y >. )  =  y ) )
3837rspcev 2897 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  T  /\  ( 2nd `  <. x ,  y
>. )  =  y
)  ->  E. z  e.  T  ( 2nd `  z )  =  y )
3933, 35, 38sylancl 643 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4039ex 423 . . . . . . 7  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y ) )
4123, 40rexlimi 2673 . . . . . 6  |-  ( E. x  e.  A  y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4219, 41impbii 180 . . . . 5  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  <->  E. x  e.  A  y  e.  B )
43 fvelimab 5594 . . . . . 6  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( y  e.  ( 2nd " T )  <->  E. z  e.  T  ( 2nd `  z )  =  y ) )
444, 5, 43mp2an 653 . . . . 5  |-  ( y  e.  ( 2nd " T
)  <->  E. z  e.  T  ( 2nd `  z )  =  y )
45 eliun 3925 . . . . 5  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
4642, 44, 453bitr4i 268 . . . 4  |-  ( y  e.  ( 2nd " T
)  <->  y  e.  U_ x  e.  A  B
)
4746eqriv 2293 . . 3  |-  ( 2nd " T )  =  U_ x  e.  A  B
488, 47eqtr3i 2318 . 2  |-  ran  ( 2nd  |`  T )  = 
U_ x  e.  A  B
49 df-fo 5277 . 2  |-  ( ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B 
<->  ( ( 2nd  |`  T )  Fn  T  /\  ran  ( 2nd  |`  T )  =  U_ x  e.  A  B ) )
507, 48, 49mpbir2an 886 1  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165   {csn 3653   <.cop 3656   U_ciun 3921    X. cxp 4703   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271   2ndc2nd 6137
This theorem is referenced by:  iundomg  8179
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-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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  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-fo 5277  df-fv 5279  df-2nd 6139
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