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Theorem djuun 6750
Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
Assertion
Ref Expression
djuun  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )

Proof of Theorem djuun
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ima 4449 . . . 4  |-  (inl " A )  =  ran  (inl  |`  A )
2 inlresf1 6743 . . . . 5  |-  (inl  |`  A ) : A -1-1-> ( A B )
3 f1rn 5211 . . . . 5  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  ran  (inl  |`  A )  C_  ( A B ) )
42, 3ax-mp 7 . . . 4  |-  ran  (inl  |`  A )  C_  ( A B )
51, 4eqsstri 3056 . . 3  |-  (inl " A )  C_  ( A B )
6 df-ima 4449 . . . 4  |-  (inr " B )  =  ran  (inr  |`  B )
7 inrresf1 6744 . . . . 5  |-  (inr  |`  B ) : B -1-1-> ( A B )
8 f1rn 5211 . . . . 5  |-  ( (inr  |`  B ) : B -1-1-> ( A B )  ->  ran  (inr  |`  B )  C_  ( A B ) )
97, 8ax-mp 7 . . . 4  |-  ran  (inr  |`  B )  C_  ( A B )
106, 9eqsstri 3056 . . 3  |-  (inr " B )  C_  ( A B )
115, 10unssi 3175 . 2  |-  ( (inl " A )  u.  (inr " B ) )  C_  ( A B )
12 djur 6747 . . . . 5  |-  ( x  e.  ( A B )  ->  ( E. y  e.  A  x  =  (inl `  y )  \/ 
E. y  e.  B  x  =  (inr `  y
) ) )
13 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
14 djulf1o 6740 . . . . . . . . . . 11  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
15 f1odm 5251 . . . . . . . . . . 11  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  dom inl  =  _V )
1614, 15ax-mp 7 . . . . . . . . . 10  |-  dom inl  =  _V
1713, 16eleqtrri 2163 . . . . . . . . 9  |-  y  e. 
dom inl
18 simpl 107 . . . . . . . . 9  |-  ( ( y  e.  A  /\  x  =  (inl `  y
) )  ->  y  e.  A )
19 df-inl 6729 . . . . . . . . . . 11  |- inl  =  ( z  e.  _V  |->  <. (/)
,  z >. )
2019funmpt2 5047 . . . . . . . . . 10  |-  Fun inl
21 funfvima 5518 . . . . . . . . . 10  |-  ( ( Fun inl  /\  y  e.  dom inl )  ->  ( y  e.  A  ->  (inl `  y )  e.  (inl " A ) ) )
2220, 21mpan 415 . . . . . . . . 9  |-  ( y  e.  dom inl  ->  ( y  e.  A  ->  (inl `  y )  e.  (inl " A ) ) )
2317, 18, 22mpsyl 64 . . . . . . . 8  |-  ( ( y  e.  A  /\  x  =  (inl `  y
) )  ->  (inl `  y )  e.  (inl " A ) )
24 eleq1 2150 . . . . . . . . 9  |-  ( x  =  (inl `  y
)  ->  ( x  e.  (inl " A )  <-> 
(inl `  y )  e.  (inl " A ) ) )
2524adantl 271 . . . . . . . 8  |-  ( ( y  e.  A  /\  x  =  (inl `  y
) )  ->  (
x  e.  (inl " A )  <->  (inl `  y
)  e.  (inl " A ) ) )
2623, 25mpbird 165 . . . . . . 7  |-  ( ( y  e.  A  /\  x  =  (inl `  y
) )  ->  x  e.  (inl " A ) )
2726rexlimiva 2484 . . . . . 6  |-  ( E. y  e.  A  x  =  (inl `  y
)  ->  x  e.  (inl " A ) )
28 djurf1o 6741 . . . . . . . . . . 11  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
29 f1odm 5251 . . . . . . . . . . 11  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  dom inr  =  _V )
3028, 29ax-mp 7 . . . . . . . . . 10  |-  dom inr  =  _V
3113, 30eleqtrri 2163 . . . . . . . . 9  |-  y  e. 
dom inr
32 simpl 107 . . . . . . . . 9  |-  ( ( y  e.  B  /\  x  =  (inr `  y
) )  ->  y  e.  B )
33 f1ofun 5249 . . . . . . . . . . 11  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  Fun inr )
3428, 33ax-mp 7 . . . . . . . . . 10  |-  Fun inr
35 funfvima 5518 . . . . . . . . . 10  |-  ( ( Fun inr  /\  y  e.  dom inr )  ->  ( y  e.  B  ->  (inr `  y )  e.  (inr " B ) ) )
3634, 35mpan 415 . . . . . . . . 9  |-  ( y  e.  dom inr  ->  ( y  e.  B  ->  (inr `  y )  e.  (inr " B ) ) )
3731, 32, 36mpsyl 64 . . . . . . . 8  |-  ( ( y  e.  B  /\  x  =  (inr `  y
) )  ->  (inr `  y )  e.  (inr " B ) )
38 eleq1 2150 . . . . . . . . 9  |-  ( x  =  (inr `  y
)  ->  ( x  e.  (inr " B )  <-> 
(inr `  y )  e.  (inr " B ) ) )
3938adantl 271 . . . . . . . 8  |-  ( ( y  e.  B  /\  x  =  (inr `  y
) )  ->  (
x  e.  (inr " B )  <->  (inr `  y
)  e.  (inr " B ) ) )
4037, 39mpbird 165 . . . . . . 7  |-  ( ( y  e.  B  /\  x  =  (inr `  y
) )  ->  x  e.  (inr " B ) )
4140rexlimiva 2484 . . . . . 6  |-  ( E. y  e.  B  x  =  (inr `  y
)  ->  x  e.  (inr " B ) )
4227, 41orim12i 711 . . . . 5  |-  ( ( E. y  e.  A  x  =  (inl `  y
)  \/  E. y  e.  B  x  =  (inr `  y ) )  ->  ( x  e.  (inl " A )  \/  x  e.  (inr " B ) ) )
4312, 42syl 14 . . . 4  |-  ( x  e.  ( A B )  ->  ( x  e.  (inl " A )  \/  x  e.  (inr " B ) ) )
44 elun 3141 . . . 4  |-  ( x  e.  ( (inl " A )  u.  (inr " B ) )  <->  ( x  e.  (inl " A )  \/  x  e.  (inr " B ) ) )
4543, 44sylibr 132 . . 3  |-  ( x  e.  ( A B )  ->  x  e.  ( (inl " A )  u.  (inr " B
) ) )
4645ssriv 3029 . 2  |-  ( A B )  C_  (
(inl " A )  u.  (inr " B ) )
4711, 46eqssi 3041 1  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360   _Vcvv 2619    u. cun 2997    C_ wss 2999   (/)c0 3286   {csn 3444   <.cop 3447    X. cxp 4434   dom cdm 4436   ran crn 4437    |` cres 4438   "cima 4439   Fun wfun 5004   -1-1->wf1 5007   -1-1-onto->wf1o 5009   ` cfv 5010   1oc1o 6166   ⊔ cdju 6720  inlcinl 6727  inrcinr 6728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-1st 5903  df-2nd 5904  df-1o 6173  df-dju 6721  df-inl 6729  df-inr 6730
This theorem is referenced by:  dju1p1e2  6813
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