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Theorem djulclb 6726
Description: Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
Assertion
Ref Expression
djulclb  |-  ( C  e.  V  ->  ( C  e.  A  <->  (inl `  C
)  e.  ( A B ) ) )

Proof of Theorem djulclb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 djulcl 6722 . 2  |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B ) )
2 1n0 6179 . . . . . . . . . 10  |-  1o  =/=  (/)
32necomi 2340 . . . . . . . . 9  |-  (/)  =/=  1o
4 0ex 3958 . . . . . . . . . 10  |-  (/)  e.  _V
54elsn 3457 . . . . . . . . 9  |-  ( (/)  e.  { 1o }  <->  (/)  =  1o )
63, 5nemtbir 2344 . . . . . . . 8  |-  -.  (/)  e.  { 1o }
76intnanr 877 . . . . . . 7  |-  -.  ( (/) 
e.  { 1o }  /\  C  e.  B
)
8 opelxp 4457 . . . . . . 7  |-  ( <. (/)
,  C >.  e.  ( { 1o }  X.  B )  <->  ( (/)  e.  { 1o }  /\  C  e.  B ) )
97, 8mtbir 631 . . . . . 6  |-  -.  <. (/)
,  C >.  e.  ( { 1o }  X.  B )
10 elex 2630 . . . . . . . . . . . 12  |-  ( C  e.  V  ->  C  e.  _V )
11 opexg 4046 . . . . . . . . . . . . 13  |-  ( (
(/)  e.  _V  /\  C  e.  V )  ->  <. (/) ,  C >.  e.  _V )
124, 11mpan 415 . . . . . . . . . . . 12  |-  ( C  e.  V  ->  <. (/) ,  C >.  e.  _V )
13 opeq2 3618 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  <. (/) ,  x >.  =  <. (/) ,  C >. )
14 df-inl 6718 . . . . . . . . . . . . 13  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
1513, 14fvmptg 5364 . . . . . . . . . . . 12  |-  ( ( C  e.  _V  /\  <. (/)
,  C >.  e.  _V )  ->  (inl `  C
)  =  <. (/) ,  C >. )
1610, 12, 15syl2anc 403 . . . . . . . . . . 11  |-  ( C  e.  V  ->  (inl `  C )  =  <. (/)
,  C >. )
1716adantr 270 . . . . . . . . . 10  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  (inl `  C
)  =  <. (/) ,  C >. )
18 df-dju 6710 . . . . . . . . . . . . 13  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
1918eleq2i 2154 . . . . . . . . . . . 12  |-  ( (inl
`  C )  e.  ( A B )  <->  (inl
`  C )  e.  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) ) )
2019biimpi 118 . . . . . . . . . . 11  |-  ( (inl
`  C )  e.  ( A B )  ->  (inl `  C )  e.  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) ) )
2120adantl 271 . . . . . . . . . 10  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  (inl `  C
)  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
2217, 21eqeltrrd 2165 . . . . . . . . 9  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  <. (/) ,  C >.  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
23 elun 3139 . . . . . . . . 9  |-  ( <. (/)
,  C >.  e.  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )  <->  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  \/  <. (/) ,  C >.  e.  ( { 1o }  X.  B ) ) )
2422, 23sylib 120 . . . . . . . 8  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  \/  <. (/) ,  C >.  e.  ( { 1o }  X.  B ) ) )
2524orcomd 683 . . . . . . 7  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( <. (/)
,  C >.  e.  ( { 1o }  X.  B )  \/  <. (/)
,  C >.  e.  ( { (/) }  X.  A
) ) )
2625ord 678 . . . . . 6  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( -.  <. (/)
,  C >.  e.  ( { 1o }  X.  B )  ->  <. (/) ,  C >.  e.  ( { (/) }  X.  A ) ) )
279, 26mpi 15 . . . . 5  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  <. (/) ,  C >.  e.  ( { (/) }  X.  A ) )
28 opelxp 4457 . . . . 5  |-  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  <->  ( (/)  e.  { (/)
}  /\  C  e.  A ) )
2927, 28sylib 120 . . . 4  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( (/)  e.  { (/)
}  /\  C  e.  A ) )
3029simprd 112 . . 3  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  C  e.  A )
3130ex 113 . 2  |-  ( C  e.  V  ->  (
(inl `  C )  e.  ( A B )  ->  C  e.  A ) )
321, 31impbid2 141 1  |-  ( C  e.  V  ->  ( C  e.  A  <->  (inl `  C
)  e.  ( A B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2995   (/)c0 3284   {csn 3441   <.cop 3444    X. cxp 4426   ` cfv 5002   1oc1o 6156   ⊔ cdju 6709  inlcinl 6716
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 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027
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-ne 2256  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-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-1o 6163  df-dju 6710  df-inl 6718
This theorem is referenced by:  exmidfodomrlemr  6807
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