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Theorem djulcl 6743
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djulcl  |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B ) )

Proof of Theorem djulcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2630 . . 3  |-  ( C  e.  A  ->  C  e.  _V )
2 0ex 3966 . . . . 5  |-  (/)  e.  _V
32snid 3475 . . . 4  |-  (/)  e.  { (/)
}
4 opelxpi 4469 . . . 4  |-  ( (
(/)  e.  { (/) }  /\  C  e.  A )  -> 
<. (/) ,  C >.  e.  ( { (/) }  X.  A ) )
53, 4mpan 415 . . 3  |-  ( C  e.  A  ->  <. (/) ,  C >.  e.  ( { (/) }  X.  A ) )
6 opeq2 3623 . . . 4  |-  ( x  =  C  ->  <. (/) ,  x >.  =  <. (/) ,  C >. )
7 df-inl 6739 . . . 4  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
86, 7fvmptg 5380 . . 3  |-  ( ( C  e.  _V  /\  <. (/)
,  C >.  e.  ( { (/) }  X.  A
) )  ->  (inl `  C )  =  <. (/)
,  C >. )
91, 5, 8syl2anc 403 . 2  |-  ( C  e.  A  ->  (inl `  C )  =  <. (/)
,  C >. )
10 elun1 3167 . . . 4  |-  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  ->  <. (/) ,  C >.  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
115, 10syl 14 . . 3  |-  ( C  e.  A  ->  <. (/) ,  C >.  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
12 df-dju 6731 . . 3  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
1311, 12syl6eleqr 2181 . 2  |-  ( C  e.  A  ->  <. (/) ,  C >.  e.  ( A B ) )
149, 13eqeltrd 2164 1  |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2997   (/)c0 3286   {csn 3446   <.cop 3449    X. cxp 4436   ` cfv 5015   1oc1o 6174   ⊔ cdju 6730  inlcinl 6737
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036
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 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-dju 6731  df-inl 6739
This theorem is referenced by:  djulclb  6747  updjudhcoinlf  6771  fodjuomnilem0  6802  exmidfodomrlemr  6828  exmidfodomrlemrALT  6829
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