MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunfo Unicode version

Theorem iunfo 8161
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 6140 . . . 4  |-  2nd : _V -onto-> _V
2 fof 5451 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
3 ffn 5389 . . . 4  |-  ( 2nd
: _V --> _V  ->  2nd 
Fn  _V )
41, 2, 3mp2b 9 . . 3  |-  2nd  Fn  _V
5 ssv 3198 . . 3  |-  T  C_  _V
6 fnssres 5357 . . 3  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( 2nd  |`  T )  Fn  T )
74, 5, 6mp2an 653 . 2  |-  ( 2nd  |`  T )  Fn  T
8 df-ima 4702 . . 3  |-  ( 2nd " T )  =  ran  ( 2nd  |`  T )
9 iunfo.1 . . . . . . . . . . 11  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
109eleq2i 2347 . . . . . . . . . 10  |-  ( z  e.  T  <->  z  e.  U_ x  e.  A  ( { x }  X.  B ) )
11 eliun 3909 . . . . . . . . . 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 6150 . . . . . . . . . . 11  |-  ( z  e.  ( { x }  X.  B )  -> 
( 2nd `  z
)  e.  B )
14 eleq1 2343 . . . . . . . . . . 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 2654 . . . . . . . . 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 2662 . . . . . 6  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  ->  E. x  e.  A  y  e.  B )
20 nfiu1 3933 . . . . . . . . 9  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
219, 20nfcxfr 2416 . . . . . . . 8  |-  F/_ x T
22 nfv 1605 . . . . . . . 8  |-  F/ x
( 2nd `  z
)  =  y
2321, 22nfrex 2598 . . . . . . 7  |-  F/ x E. z  e.  T  ( 2nd `  z )  =  y
24 ssiun2 3945 . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2827snid 3667 . . . . . . . . . . . . 13  |-  x  e. 
{ x }
29 opelxp 4719 . . . . . . . . . . . . 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 3181 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B ) )
3332, 9syl6eleqr 2374 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  T )
34 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
3527, 34op2nd 6129 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
36 fveq2 5525 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  ( 2nd `  <. x ,  y
>. ) )
3736eqeq1d 2291 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  z )  =  y  <-> 
( 2nd `  <. x ,  y >. )  =  y ) )
3837rspcev 2884 . . . . . . . . 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 2660 . . . . . 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 5578 . . . . . 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 3909 . . . . 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 2280 . . 3  |-  ( 2nd " T )  =  U_ x  e.  A  B
488, 47eqtr3i 2305 . 2  |-  ran  ( 2nd  |`  T )  = 
U_ x  e.  A  B
49 df-fo 5261 . 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 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255   2ndc2nd 6121
This theorem is referenced by:  iundomg  8163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-2nd 6123
  Copyright terms: Public domain W3C validator