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Theorem djulclb 7053
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 7049 . 2  |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B ) )
2 1n0 6432 . . . . . . . . . 10  |-  1o  =/=  (/)
32necomi 2432 . . . . . . . . 9  |-  (/)  =/=  1o
4 0ex 4130 . . . . . . . . . 10  |-  (/)  e.  _V
54elsn 3608 . . . . . . . . 9  |-  ( (/)  e.  { 1o }  <->  (/)  =  1o )
63, 5nemtbir 2436 . . . . . . . 8  |-  -.  (/)  e.  { 1o }
76intnanr 930 . . . . . . 7  |-  -.  ( (/) 
e.  { 1o }  /\  C  e.  B
)
8 opelxp 4656 . . . . . . 7  |-  ( <. (/)
,  C >.  e.  ( { 1o }  X.  B )  <->  ( (/)  e.  { 1o }  /\  C  e.  B ) )
97, 8mtbir 671 . . . . . 6  |-  -.  <. (/)
,  C >.  e.  ( { 1o }  X.  B )
10 elex 2748 . . . . . . . . . . . 12  |-  ( C  e.  V  ->  C  e.  _V )
11 opexg 4228 . . . . . . . . . . . . 13  |-  ( (
(/)  e.  _V  /\  C  e.  V )  ->  <. (/) ,  C >.  e.  _V )
124, 11mpan 424 . . . . . . . . . . . 12  |-  ( C  e.  V  ->  <. (/) ,  C >.  e.  _V )
13 opeq2 3779 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  <. (/) ,  x >.  =  <. (/) ,  C >. )
14 df-inl 7045 . . . . . . . . . . . . 13  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
1513, 14fvmptg 5592 . . . . . . . . . . . 12  |-  ( ( C  e.  _V  /\  <. (/)
,  C >.  e.  _V )  ->  (inl `  C
)  =  <. (/) ,  C >. )
1610, 12, 15syl2anc 411 . . . . . . . . . . 11  |-  ( C  e.  V  ->  (inl `  C )  =  <. (/)
,  C >. )
1716adantr 276 . . . . . . . . . 10  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  (inl `  C
)  =  <. (/) ,  C >. )
18 df-dju 7036 . . . . . . . . . . . . 13  |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )
1918eleq2i 2244 . . . . . . . . . . . 12  |-  ( (inl
`  C )  e.  ( A B )  <->  (inl
`  C )  e.  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) ) )
2019biimpi 120 . . . . . . . . . . 11  |-  ( (inl
`  C )  e.  ( A B )  ->  (inl `  C )  e.  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) ) )
2120adantl 277 . . . . . . . . . 10  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  (inl `  C
)  e.  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
2217, 21eqeltrrd 2255 . . . . . . . . 9  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  <. (/) ,  C >.  e.  ( ( {
(/) }  X.  A
)  u.  ( { 1o }  X.  B
) ) )
23 elun 3276 . . . . . . . . 9  |-  ( <. (/)
,  C >.  e.  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B
) )  <->  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  \/  <. (/) ,  C >.  e.  ( { 1o }  X.  B ) ) )
2422, 23sylib 122 . . . . . . . 8  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  \/  <. (/) ,  C >.  e.  ( { 1o }  X.  B ) ) )
2524orcomd 729 . . . . . . 7  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( <. (/)
,  C >.  e.  ( { 1o }  X.  B )  \/  <. (/)
,  C >.  e.  ( { (/) }  X.  A
) ) )
2625ord 724 . . . . . 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 4656 . . . . 5  |-  ( <. (/)
,  C >.  e.  ( { (/) }  X.  A
)  <->  ( (/)  e.  { (/)
}  /\  C  e.  A ) )
2927, 28sylib 122 . . . 4  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  ( (/)  e.  { (/)
}  /\  C  e.  A ) )
3029simprd 114 . . 3  |-  ( ( C  e.  V  /\  (inl `  C )  e.  ( A B )
)  ->  C  e.  A )
3130ex 115 . 2  |-  ( C  e.  V  ->  (
(inl `  C )  e.  ( A B )  ->  C  e.  A ) )
321, 31impbid2 143 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 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   _Vcvv 2737    u. cun 3127   (/)c0 3422   {csn 3592   <.cop 3595    X. cxp 4624   ` cfv 5216   1oc1o 6409   ⊔ cdju 7035  inlcinl 7043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-1o 6416  df-dju 7036  df-inl 7045
This theorem is referenced by:  exmidfodomrlemr  7200
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