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Theorem djulclr 6941
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
Assertion
Ref Expression
djulclr  |-  ( C  e.  A  ->  (
(inl  |`  A ) `  C )  e.  ( A B ) )

Proof of Theorem djulclr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvres 5452 . 2  |-  ( C  e.  A  ->  (
(inl  |`  A ) `  C )  =  (inl
`  C ) )
2 elex 2700 . . . 4  |-  ( C  e.  A  ->  C  e.  _V )
3 0ex 4062 . . . . . 6  |-  (/)  e.  _V
43snid 3562 . . . . 5  |-  (/)  e.  { (/)
}
5 opelxpi 4578 . . . . 5  |-  ( (
(/)  e.  { (/) }  /\  C  e.  A )  -> 
<. (/) ,  C >.  e.  ( { (/) }  X.  A ) )
64, 5mpan 421 . . . 4  |-  ( C  e.  A  ->  <. (/) ,  C >.  e.  ( { (/) }  X.  A ) )
7 opeq2 3713 . . . . 5  |-  ( x  =  C  ->  <. (/) ,  x >.  =  <. (/) ,  C >. )
8 df-inl 6939 . . . . 5  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
97, 8fvmptg 5504 . . . 4  |-  ( ( C  e.  _V  /\  <. (/)
,  C >.  e.  ( { (/) }  X.  A
) )  ->  (inl `  C )  =  <. (/)
,  C >. )
102, 6, 9syl2anc 409 . . 3  |-  ( C  e.  A  ->  (inl `  C )  =  <. (/)
,  C >. )
11 elun1 3247 . . . . 5  |-  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  ->  <. (/) ,  C >.  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
126, 11syl 14 . . . 4  |-  ( C  e.  A  ->  <. (/) ,  C >.  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
13 df-dju 6930 . . . 4  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
1412, 13eleqtrrdi 2234 . . 3  |-  ( C  e.  A  ->  <. (/) ,  C >.  e.  ( A B ) )
1510, 14eqeltrd 2217 . 2  |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B ) )
161, 15eqeltrd 2217 1  |-  ( C  e.  A  ->  (
(inl  |`  A ) `  C )  e.  ( A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481   _Vcvv 2689    u. cun 3073   (/)c0 3367   {csn 3531   <.cop 3534    X. cxp 4544    |` cres 4548   ` cfv 5130   1oc1o 6313   ⊔ cdju 6929  inlcinl 6937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-res 4558  df-iota 5095  df-fun 5132  df-fv 5138  df-dju 6930  df-inl 6939
This theorem is referenced by:  inlresf1  6953
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