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Theorem iunfo 8804
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 6697 . . . 4  |-  2nd : _V -onto-> _V
2 fof 5718 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
3 ffn 5657 . . . 4  |-  ( 2nd
: _V --> _V  ->  2nd 
Fn  _V )
41, 2, 3mp2b 10 . . 3  |-  2nd  Fn  _V
5 ssv 3474 . . 3  |-  T  C_  _V
6 fnssres 5622 . . 3  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( 2nd  |`  T )  Fn  T )
74, 5, 6mp2an 672 . 2  |-  ( 2nd  |`  T )  Fn  T
8 df-ima 4951 . . 3  |-  ( 2nd " T )  =  ran  ( 2nd  |`  T )
9 iunfo.1 . . . . . . . . . . 11  |-  T  = 
U_ x  e.  A  ( { x }  X.  B )
109eleq2i 2529 . . . . . . . . . 10  |-  ( z  e.  T  <->  z  e.  U_ x  e.  A  ( { x }  X.  B ) )
11 eliun 4273 . . . . . . . . . 10  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
1210, 11bitri 249 . . . . . . . . 9  |-  ( z  e.  T  <->  E. x  e.  A  z  e.  ( { x }  X.  B ) )
13 xp2nd 6707 . . . . . . . . . . 11  |-  ( z  e.  ( { x }  X.  B )  -> 
( 2nd `  z
)  e.  B )
14 eleq1 2523 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  y  ->  (
( 2nd `  z
)  e.  B  <->  y  e.  B ) )
1513, 14syl5ib 219 . . . . . . . . . 10  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  ( { x }  X.  B
)  ->  y  e.  B ) )
1615reximdv 2923 . . . . . . . . 9  |-  ( ( 2nd `  z )  =  y  ->  ( E. x  e.  A  z  e.  ( {
x }  X.  B
)  ->  E. x  e.  A  y  e.  B ) )
1712, 16syl5bi 217 . . . . . . . 8  |-  ( ( 2nd `  z )  =  y  ->  (
z  e.  T  ->  E. x  e.  A  y  e.  B )
)
1817impcom 430 . . . . . . 7  |-  ( ( z  e.  T  /\  ( 2nd `  z )  =  y )  ->  E. x  e.  A  y  e.  B )
1918rexlimiva 2932 . . . . . 6  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  ->  E. x  e.  A  y  e.  B )
20 nfiu1 4298 . . . . . . . . 9  |-  F/_ x U_ x  e.  A  ( { x }  X.  B )
219, 20nfcxfr 2611 . . . . . . . 8  |-  F/_ x T
22 nfv 1674 . . . . . . . 8  |-  F/ x
( 2nd `  z
)  =  y
2321, 22nfrex 2880 . . . . . . 7  |-  F/ x E. z  e.  T  ( 2nd `  z )  =  y
24 ssiun2 4311 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
2524adantr 465 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  X.  B )  C_  U_ x  e.  A  ( {
x }  X.  B
) )
26 simpr 461 . . . . . . . . . . . 12  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  B )
27 ssnid 4004 . . . . . . . . . . . . 13  |-  x  e. 
{ x }
28 opelxp 4967 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  ( x  e.  { x }  /\  y  e.  B )
)
2927, 28mpbiran 909 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( { x }  X.  B )  <->  y  e.  B )
3026, 29sylibr 212 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( { x }  X.  B ) )
3125, 30sseldd 3455 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  U_ x  e.  A  ( { x }  X.  B ) )
3231, 9syl6eleqr 2550 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  T )
33 vex 3071 . . . . . . . . . 10  |-  x  e. 
_V
34 vex 3071 . . . . . . . . . 10  |-  y  e. 
_V
3533, 34op2nd 6686 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
36 fveq2 5789 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  ( 2nd `  <. x ,  y
>. ) )
3736eqeq1d 2453 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( ( 2nd `  z )  =  y  <-> 
( 2nd `  <. x ,  y >. )  =  y ) )
3837rspcev 3169 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  T  /\  ( 2nd `  <. x ,  y
>. )  =  y
)  ->  E. z  e.  T  ( 2nd `  z )  =  y )
3932, 35, 38sylancl 662 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4039ex 434 . . . . . . 7  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y ) )
4123, 40rexlimi 2930 . . . . . 6  |-  ( E. x  e.  A  y  e.  B  ->  E. z  e.  T  ( 2nd `  z )  =  y )
4219, 41impbii 188 . . . . 5  |-  ( E. z  e.  T  ( 2nd `  z )  =  y  <->  E. x  e.  A  y  e.  B )
43 fvelimab 5846 . . . . . 6  |-  ( ( 2nd  Fn  _V  /\  T  C_  _V )  -> 
( y  e.  ( 2nd " T )  <->  E. z  e.  T  ( 2nd `  z )  =  y ) )
444, 5, 43mp2an 672 . . . . 5  |-  ( y  e.  ( 2nd " T
)  <->  E. z  e.  T  ( 2nd `  z )  =  y )
45 eliun 4273 . . . . 5  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
4642, 44, 453bitr4i 277 . . . 4  |-  ( y  e.  ( 2nd " T
)  <->  y  e.  U_ x  e.  A  B
)
4746eqriv 2447 . . 3  |-  ( 2nd " T )  =  U_ x  e.  A  B
488, 47eqtr3i 2482 . 2  |-  ran  ( 2nd  |`  T )  = 
U_ x  e.  A  B
49 df-fo 5522 . 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 911 1  |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3068    C_ wss 3426   {csn 3975   <.cop 3981   U_ciun 4269    X. cxp 4936   ran crn 4939    |` cres 4940   "cima 4941    Fn wfn 5511   -->wf 5512   -onto->wfo 5514   ` cfv 5516   2ndc2nd 6676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fo 5522  df-fv 5524  df-2nd 6678
This theorem is referenced by:  iundomg  8806
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