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Theorem iunfo 8419
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 6370 . . . 4  |-  2nd : _V -onto-> _V
2 fof 5656 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
3 ffn 5594 . . . 4  |-  ( 2nd
: _V --> _V  ->  2nd 
Fn  _V )
41, 2, 3mp2b 10 . . 3  |-  2nd  Fn  _V
5 ssv 3370 . . 3  |-  T  C_  _V
6 fnssres 5561 . . 3  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( 2nd  |`  T )  Fn  T )
74, 5, 6mp2an 655 . 2  |-  ( 2nd  |`  T )  Fn  T
8 df-ima 4894 . . 3  |-  ( 2nd " T )  =  ran  ( 2nd  |`  T )
9 iunfo.1 . . . . . . . . . . 11  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
109eleq2i 2502 . . . . . . . . . 10  |-  ( z  e.  T  <->  z  e.  U_ x  e.  A  ( { x }  X.  B ) )
11 eliun 4099 . . . . . . . . . 10  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
1210, 11bitri 242 . . . . . . . . 9  |-  ( z  e.  T  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
13 xp2nd 6380 . . . . . . . . . . 11  |-  ( z  e.  ( { x }  X.  B )  -> 
( 2nd `  z
)  e.  B )
14 eleq1 2498 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  y  ->  (
( 2nd `  z
)  e.  B  <->  y  e.  B ) )
1513, 14syl5ib 212 . . . . . . . . . 10  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  ( { x }  X.  B
)  ->  y  e.  B ) )
1615reximdv 2819 . . . . . . . . 9  |-  ( ( 2nd `  z )  =  y  ->  ( E. x  e.  A  z  e.  ( {
x }  X.  B
)  ->  E. x  e.  A  y  e.  B ) )
1712, 16syl5bi 210 . . . . . . . 8  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  T  ->  E. x  e.  A  y  e.  B )
)
1817impcom 421 . . . . . . 7  |-  ( ( z  e.  T  /\  ( 2nd `  z )  =  y )  ->  E. x  e.  A  y  e.  B )
1918rexlimiva 2827 . . . . . 6  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  ->  E. x  e.  A  y  e.  B )
20 nfiu1 4123 . . . . . . . . 9  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
219, 20nfcxfr 2571 . . . . . . . 8  |-  F/_ x T
22 nfv 1630 . . . . . . . 8  |-  F/ x
( 2nd `  z
)  =  y
2321, 22nfrex 2763 . . . . . . 7  |-  F/ x E. z  e.  T  ( 2nd `  z )  =  y
24 ssiun2 4136 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
2524adantr 453 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
26 simpr 449 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  B )
27 vex 2961 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2827snid 3843 . . . . . . . . . . . . 13  |-  x  e. 
{ x }
29 opelxp 4911 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  ( x  e.  { x }  /\  y  e.  B )
)
3028, 29mpbiran 886 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  y  e.  B )
3126, 30sylibr 205 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( { x }  X.  B ) )
3225, 31sseldd 3351 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B ) )
3332, 9syl6eleqr 2529 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  T )
34 vex 2961 . . . . . . . . . 10  |-  y  e. 
_V
3527, 34op2nd 6359 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
36 fveq2 5731 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  ( 2nd `  <. x ,  y
>. ) )
3736eqeq1d 2446 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  z )  =  y  <-> 
( 2nd `  <. x ,  y >. )  =  y ) )
3837rspcev 3054 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  T  /\  ( 2nd `  <. x ,  y
>. )  =  y
)  ->  E. z  e.  T  ( 2nd `  z )  =  y )
3933, 35, 38sylancl 645 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4039ex 425 . . . . . . 7  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y ) )
4123, 40rexlimi 2825 . . . . . 6  |-  ( E. x  e.  A  y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4219, 41impbii 182 . . . . 5  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  <->  E. x  e.  A  y  e.  B )
43 fvelimab 5785 . . . . . 6  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( y  e.  ( 2nd " T )  <->  E. z  e.  T  ( 2nd `  z )  =  y ) )
444, 5, 43mp2an 655 . . . . 5  |-  ( y  e.  ( 2nd " T
)  <->  E. z  e.  T  ( 2nd `  z )  =  y )
45 eliun 4099 . . . . 5  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
4642, 44, 453bitr4i 270 . . . 4  |-  ( y  e.  ( 2nd " T
)  <->  y  e.  U_ x  e.  A  B
)
4746eqriv 2435 . . 3  |-  ( 2nd " T )  =  U_ x  e.  A  B
488, 47eqtr3i 2460 . 2  |-  ran  ( 2nd  |`  T )  = 
U_ x  e.  A  B
49 df-fo 5463 . 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 888 1  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958    C_ wss 3322   {csn 3816   <.cop 3819   U_ciun 4095    X. cxp 4879   ran crn 4882    |` cres 4883   "cima 4884    Fn wfn 5452   -->wf 5453   -onto->wfo 5455   ` cfv 5457   2ndc2nd 6351
This theorem is referenced by:  iundomg  8421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-2nd 6353
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