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Theorem djur 6736
Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
djur  |-  ( C  e.  ( A B )  ->  ( E. x  e.  A  C  =  (inl `  x )  \/ 
E. x  e.  B  C  =  (inr `  x
) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem djur
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dju 6710 . . . 4  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
21eleq2i 2154 . . 3  |-  ( C  e.  ( A B )  <-> 
C  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
3 elun 3139 . . 3  |-  ( C  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) )  <->  ( C  e.  ( { (/) }  X.  A )  \/  C  e.  ( { 1o }  X.  B ) ) )
42, 3sylbb 121 . 2  |-  ( C  e.  ( A B )  ->  ( C  e.  ( { (/) }  X.  A )  \/  C  e.  ( { 1o }  X.  B ) ) )
5 xp2nd 5919 . . . 4  |-  ( C  e.  ( { (/) }  X.  A )  -> 
( 2nd `  C
)  e.  A )
6 1st2nd2 5927 . . . . . 6  |-  ( C  e.  ( { (/) }  X.  A )  ->  C  =  <. ( 1st `  C ) ,  ( 2nd `  C )
>. )
7 xp1st 5918 . . . . . . 7  |-  ( C  e.  ( { (/) }  X.  A )  -> 
( 1st `  C
)  e.  { (/) } )
8 elsni 3459 . . . . . . 7  |-  ( ( 1st `  C )  e.  { (/) }  ->  ( 1st `  C )  =  (/) )
9 opeq1 3617 . . . . . . . 8  |-  ( ( 1st `  C )  =  (/)  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  =  <. (/) ,  ( 2nd `  C )
>. )
109eqeq2d 2099 . . . . . . 7  |-  ( ( 1st `  C )  =  (/)  ->  ( C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >.  <->  C  =  <.
(/) ,  ( 2nd `  C ) >. )
)
117, 8, 103syl 17 . . . . . 6  |-  ( C  e.  ( { (/) }  X.  A )  -> 
( C  =  <. ( 1st `  C ) ,  ( 2nd `  C
) >. 
<->  C  =  <. (/) ,  ( 2nd `  C )
>. ) )
126, 11mpbid 145 . . . . 5  |-  ( C  e.  ( { (/) }  X.  A )  ->  C  =  <. (/) ,  ( 2nd `  C )
>. )
13 2ndexg 5921 . . . . . 6  |-  ( C  e.  ( { (/) }  X.  A )  -> 
( 2nd `  C
)  e.  _V )
14 0ex 3958 . . . . . . 7  |-  (/)  e.  _V
15 opexg 4046 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  ( 2nd `  C )  e. 
_V )  ->  <. (/) ,  ( 2nd `  C )
>.  e.  _V )
1614, 13, 15sylancr 405 . . . . . 6  |-  ( C  e.  ( { (/) }  X.  A )  ->  <.
(/) ,  ( 2nd `  C ) >.  e.  _V )
17 opeq2 3618 . . . . . . 7  |-  ( y  =  ( 2nd `  C
)  ->  <. (/) ,  y
>.  =  <. (/) ,  ( 2nd `  C )
>. )
18 df-inl 6718 . . . . . . 7  |- inl  =  ( y  e.  _V  |->  <. (/)
,  y >. )
1917, 18fvmptg 5364 . . . . . 6  |-  ( ( ( 2nd `  C
)  e.  _V  /\  <. (/)
,  ( 2nd `  C
) >.  e.  _V )  ->  (inl `  ( 2nd `  C ) )  = 
<. (/) ,  ( 2nd `  C ) >. )
2013, 16, 19syl2anc 403 . . . . 5  |-  ( C  e.  ( { (/) }  X.  A )  -> 
(inl `  ( 2nd `  C ) )  = 
<. (/) ,  ( 2nd `  C ) >. )
2112, 20eqtr4d 2123 . . . 4  |-  ( C  e.  ( { (/) }  X.  A )  ->  C  =  (inl `  ( 2nd `  C ) ) )
22 fveq2 5289 . . . . . 6  |-  ( x  =  ( 2nd `  C
)  ->  (inl `  x
)  =  (inl `  ( 2nd `  C ) ) )
2322eqeq2d 2099 . . . . 5  |-  ( x  =  ( 2nd `  C
)  ->  ( C  =  (inl `  x )  <->  C  =  (inl `  ( 2nd `  C ) ) ) )
2423rspcev 2722 . . . 4  |-  ( ( ( 2nd `  C
)  e.  A  /\  C  =  (inl `  ( 2nd `  C ) ) )  ->  E. x  e.  A  C  =  (inl `  x ) )
255, 21, 24syl2anc 403 . . 3  |-  ( C  e.  ( { (/) }  X.  A )  ->  E. x  e.  A  C  =  (inl `  x
) )
26 xp2nd 5919 . . . 4  |-  ( C  e.  ( { 1o }  X.  B )  -> 
( 2nd `  C
)  e.  B )
27 1st2nd2 5927 . . . . . 6  |-  ( C  e.  ( { 1o }  X.  B )  ->  C  =  <. ( 1st `  C ) ,  ( 2nd `  C )
>. )
28 xp1st 5918 . . . . . . 7  |-  ( C  e.  ( { 1o }  X.  B )  -> 
( 1st `  C
)  e.  { 1o } )
29 elsni 3459 . . . . . . 7  |-  ( ( 1st `  C )  e.  { 1o }  ->  ( 1st `  C
)  =  1o )
30 opeq1 3617 . . . . . . . 8  |-  ( ( 1st `  C )  =  1o  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  =  <. 1o , 
( 2nd `  C
) >. )
3130eqeq2d 2099 . . . . . . 7  |-  ( ( 1st `  C )  =  1o  ->  ( C  =  <. ( 1st `  C ) ,  ( 2nd `  C )
>. 
<->  C  =  <. 1o , 
( 2nd `  C
) >. ) )
3228, 29, 313syl 17 . . . . . 6  |-  ( C  e.  ( { 1o }  X.  B )  -> 
( C  =  <. ( 1st `  C ) ,  ( 2nd `  C
) >. 
<->  C  =  <. 1o , 
( 2nd `  C
) >. ) )
3327, 32mpbid 145 . . . . 5  |-  ( C  e.  ( { 1o }  X.  B )  ->  C  =  <. 1o , 
( 2nd `  C
) >. )
34 2ndexg 5921 . . . . . 6  |-  ( C  e.  ( { 1o }  X.  B )  -> 
( 2nd `  C
)  e.  _V )
35 1onn 6259 . . . . . . 7  |-  1o  e.  om
36 opexg 4046 . . . . . . 7  |-  ( ( 1o  e.  om  /\  ( 2nd `  C )  e.  _V )  ->  <. 1o ,  ( 2nd `  C ) >.  e.  _V )
3735, 34, 36sylancr 405 . . . . . 6  |-  ( C  e.  ( { 1o }  X.  B )  ->  <. 1o ,  ( 2nd `  C ) >.  e.  _V )
38 opeq2 3618 . . . . . . 7  |-  ( z  =  ( 2nd `  C
)  ->  <. 1o , 
z >.  =  <. 1o , 
( 2nd `  C
) >. )
39 df-inr 6719 . . . . . . 7  |- inr  =  ( z  e.  _V  |->  <. 1o ,  z >. )
4038, 39fvmptg 5364 . . . . . 6  |-  ( ( ( 2nd `  C
)  e.  _V  /\  <. 1o ,  ( 2nd `  C ) >.  e.  _V )  ->  (inr `  ( 2nd `  C ) )  =  <. 1o ,  ( 2nd `  C )
>. )
4134, 37, 40syl2anc 403 . . . . 5  |-  ( C  e.  ( { 1o }  X.  B )  -> 
(inr `  ( 2nd `  C ) )  = 
<. 1o ,  ( 2nd `  C ) >. )
4233, 41eqtr4d 2123 . . . 4  |-  ( C  e.  ( { 1o }  X.  B )  ->  C  =  (inr `  ( 2nd `  C ) ) )
43 fveq2 5289 . . . . . 6  |-  ( x  =  ( 2nd `  C
)  ->  (inr `  x
)  =  (inr `  ( 2nd `  C ) ) )
4443eqeq2d 2099 . . . . 5  |-  ( x  =  ( 2nd `  C
)  ->  ( C  =  (inr `  x )  <->  C  =  (inr `  ( 2nd `  C ) ) ) )
4544rspcev 2722 . . . 4  |-  ( ( ( 2nd `  C
)  e.  B  /\  C  =  (inr `  ( 2nd `  C ) ) )  ->  E. x  e.  B  C  =  (inr `  x ) )
4626, 42, 45syl2anc 403 . . 3  |-  ( C  e.  ( { 1o }  X.  B )  ->  E. x  e.  B  C  =  (inr `  x
) )
4725, 46orim12i 711 . 2  |-  ( ( C  e.  ( {
(/) }  X.  A
)  \/  C  e.  ( { 1o }  X.  B ) )  -> 
( E. x  e.  A  C  =  (inl
`  x )  \/ 
E. x  e.  B  C  =  (inr `  x
) ) )
484, 47syl 14 1  |-  ( C  e.  ( A B )  ->  ( E. x  e.  A  C  =  (inl `  x )  \/ 
E. x  e.  B  C  =  (inr `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360   _Vcvv 2619    u. cun 2995   (/)c0 3284   {csn 3441   <.cop 3444   omcom 4395    X. cxp 4426   ` cfv 5002   1stc1st 5891   2ndc2nd 5892   1oc1o 6156   ⊔ cdju 6709  inlcinl 6716  inrcinr 6717
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 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
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 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719
This theorem is referenced by:  djuun  6739  djuss  6740  updjud  6752  fodjuomnilemdc  6778  exmidfodomrlemeldju  6804
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