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Theorem iunfo 8406
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 6359 . . . 4  |-  2nd : _V -onto-> _V
2 fof 5645 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
3 ffn 5583 . . . 4  |-  ( 2nd
: _V --> _V  ->  2nd 
Fn  _V )
41, 2, 3mp2b 10 . . 3  |-  2nd  Fn  _V
5 ssv 3360 . . 3  |-  T  C_  _V
6 fnssres 5550 . . 3  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( 2nd  |`  T )  Fn  T )
74, 5, 6mp2an 654 . 2  |-  ( 2nd  |`  T )  Fn  T
8 df-ima 4883 . . 3  |-  ( 2nd " T )  =  ran  ( 2nd  |`  T )
9 iunfo.1 . . . . . . . . . . 11  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
109eleq2i 2499 . . . . . . . . . 10  |-  ( z  e.  T  <->  z  e.  U_ x  e.  A  ( { x }  X.  B ) )
11 eliun 4089 . . . . . . . . . 10  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
1210, 11bitri 241 . . . . . . . . 9  |-  ( z  e.  T  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
13 xp2nd 6369 . . . . . . . . . . 11  |-  ( z  e.  ( { x }  X.  B )  -> 
( 2nd `  z
)  e.  B )
14 eleq1 2495 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  y  ->  (
( 2nd `  z
)  e.  B  <->  y  e.  B ) )
1513, 14syl5ib 211 . . . . . . . . . 10  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  ( { x }  X.  B
)  ->  y  e.  B ) )
1615reximdv 2809 . . . . . . . . 9  |-  ( ( 2nd `  z )  =  y  ->  ( E. x  e.  A  z  e.  ( {
x }  X.  B
)  ->  E. x  e.  A  y  e.  B ) )
1712, 16syl5bi 209 . . . . . . . 8  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  T  ->  E. x  e.  A  y  e.  B )
)
1817impcom 420 . . . . . . 7  |-  ( ( z  e.  T  /\  ( 2nd `  z )  =  y )  ->  E. x  e.  A  y  e.  B )
1918rexlimiva 2817 . . . . . 6  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  ->  E. x  e.  A  y  e.  B )
20 nfiu1 4113 . . . . . . . . 9  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
219, 20nfcxfr 2568 . . . . . . . 8  |-  F/_ x T
22 nfv 1629 . . . . . . . 8  |-  F/ x
( 2nd `  z
)  =  y
2321, 22nfrex 2753 . . . . . . 7  |-  F/ x E. z  e.  T  ( 2nd `  z )  =  y
24 ssiun2 4126 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
2524adantr 452 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
26 simpr 448 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  B )
27 vex 2951 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2827snid 3833 . . . . . . . . . . . . 13  |-  x  e. 
{ x }
29 opelxp 4900 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  ( x  e.  { x }  /\  y  e.  B )
)
3028, 29mpbiran 885 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  y  e.  B )
3126, 30sylibr 204 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( { x }  X.  B ) )
3225, 31sseldd 3341 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B ) )
3332, 9syl6eleqr 2526 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  T )
34 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
3527, 34op2nd 6348 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
36 fveq2 5720 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  ( 2nd `  <. x ,  y
>. ) )
3736eqeq1d 2443 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  z )  =  y  <-> 
( 2nd `  <. x ,  y >. )  =  y ) )
3837rspcev 3044 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  T  /\  ( 2nd `  <. x ,  y
>. )  =  y
)  ->  E. z  e.  T  ( 2nd `  z )  =  y )
3933, 35, 38sylancl 644 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4039ex 424 . . . . . . 7  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y ) )
4123, 40rexlimi 2815 . . . . . 6  |-  ( E. x  e.  A  y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4219, 41impbii 181 . . . . 5  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  <->  E. x  e.  A  y  e.  B )
43 fvelimab 5774 . . . . . 6  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( y  e.  ( 2nd " T )  <->  E. z  e.  T  ( 2nd `  z )  =  y ) )
444, 5, 43mp2an 654 . . . . 5  |-  ( y  e.  ( 2nd " T
)  <->  E. z  e.  T  ( 2nd `  z )  =  y )
45 eliun 4089 . . . . 5  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
4642, 44, 453bitr4i 269 . . . 4  |-  ( y  e.  ( 2nd " T
)  <->  y  e.  U_ x  e.  A  B
)
4746eqriv 2432 . . 3  |-  ( 2nd " T )  =  U_ x  e.  A  B
488, 47eqtr3i 2457 . 2  |-  ran  ( 2nd  |`  T )  = 
U_ x  e.  A  B
49 df-fo 5452 . 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 887 1  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    C_ wss 3312   {csn 3806   <.cop 3809   U_ciun 4085    X. cxp 4868   ran crn 4871    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   ` cfv 5446   2ndc2nd 6340
This theorem is referenced by:  iundomg  8408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-2nd 6342
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