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Theorem djuun 6681
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 4417 . . . 4  |-  (inl " A )  =  ran  (inl  |`  A )
2 inlresf1 6674 . . . . 5  |-  (inl  |`  A ) : A -1-1-> ( A B )
3 f1rn 5168 . . . . 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 3042 . . 3  |-  (inl " A )  C_  ( A B )
6 df-ima 4417 . . . 4  |-  (inr " B )  =  ran  (inr  |`  B )
7 inrresf1 6675 . . . . 5  |-  (inr  |`  B ) : B -1-1-> ( A B )
8 f1rn 5168 . . . . 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 3042 . . 3  |-  (inr " B )  C_  ( A B )
115, 10unssi 3161 . 2  |-  ( (inl " A )  u.  (inr " B ) )  C_  ( A B )
12 djur 6678 . . . . 5  |-  ( x  e.  ( A B )  ->  ( E. y  e.  A  x  =  (inl `  y )  \/ 
E. y  e.  B  x  =  (inr `  y
) ) )
13 vex 2617 . . . . . . . . . 10  |-  y  e. 
_V
14 djulf1o 6671 . . . . . . . . . . 11  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
15 f1odm 5208 . . . . . . . . . . 11  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  ->  dom inl  =  _V )
1614, 15ax-mp 7 . . . . . . . . . 10  |-  dom inl  =  _V
1713, 16eleqtrri 2160 . . . . . . . . 9  |-  y  e. 
dom inl
18 simpl 107 . . . . . . . . 9  |-  ( ( y  e.  A  /\  x  =  (inl `  y
) )  ->  y  e.  A )
19 df-inl 6660 . . . . . . . . . . 11  |- inl  =  ( z  e.  _V  |->  <. (/)
,  z >. )
2019funmpt2 5009 . . . . . . . . . 10  |-  Fun inl
21 funfvima 5469 . . . . . . . . . 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 2147 . . . . . . . . 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 2480 . . . . . 6  |-  ( E. y  e.  A  x  =  (inl `  y
)  ->  x  e.  (inl " A ) )
28 djurf1o 6672 . . . . . . . . . . 11  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
29 f1odm 5208 . . . . . . . . . . 11  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  dom inr  =  _V )
3028, 29ax-mp 7 . . . . . . . . . 10  |-  dom inr  =  _V
3113, 30eleqtrri 2160 . . . . . . . . 9  |-  y  e. 
dom inr
32 simpl 107 . . . . . . . . 9  |-  ( ( y  e.  B  /\  x  =  (inr `  y
) )  ->  y  e.  B )
33 f1ofun 5206 . . . . . . . . . . 11  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  ->  Fun inr )
3428, 33ax-mp 7 . . . . . . . . . 10  |-  Fun inr
35 funfvima 5469 . . . . . . . . . 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 2147 . . . . . . . . 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 2480 . . . . . 6  |-  ( E. y  e.  B  x  =  (inr `  y
)  ->  x  e.  (inr " B ) )
4227, 41orim12i 709 . . . . 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 3127 . . . 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 3016 . 2  |-  ( A B )  C_  (
(inl " A )  u.  (inr " B ) )
4711, 46eqssi 3028 1  |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1287    e. wcel 1436   E.wrex 2356   _Vcvv 2614    u. cun 2984    C_ wss 2986   (/)c0 3272   {csn 3425   <.cop 3428    X. cxp 4402   dom cdm 4404   ran crn 4405    |` cres 4406   "cima 4407   Fun wfun 4966   -1-1->wf1 4969   -1-1-onto->wf1o 4971   ` cfv 4972   1oc1o 6109   ⊔ cdju 6651  inlcinl 6658  inrcinr 6659
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-1st 5849  df-2nd 5850  df-1o 6116  df-dju 6652  df-inl 6660  df-inr 6661
This theorem is referenced by:  dju1p1e2  6744
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