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Theorem eldju 6960
Description: Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
eldju  |-  ( C  e.  ( A B )  <-> 
( E. x  e.  A  C  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `
 x ) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem eldju
StepHypRef Expression
1 djuunr 6958 . . . 4  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
21eqcomi 2144 . . 3  |-  ( A B )  =  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )
32eleq2i 2207 . 2  |-  ( C  e.  ( A B )  <-> 
C  e.  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) ) )
4 elun 3221 . . 3  |-  ( C  e.  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  <-> 
( C  e.  ran  (inl  |`  A )  \/  C  e.  ran  (inr  |`  B ) ) )
5 djulf1or 6948 . . . . . 6  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
6 f1ofn 5375 . . . . . 6  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  (inl  |`  A )  Fn  A )
7 fvelrnb 5476 . . . . . 6  |-  ( (inl  |`  A )  Fn  A  ->  ( C  e.  ran  (inl  |`  A )  <->  E. x  e.  A  ( (inl  |`  A ) `  x
)  =  C ) )
85, 6, 7mp2b 8 . . . . 5  |-  ( C  e.  ran  (inl  |`  A )  <->  E. x  e.  A  ( (inl  |`  A ) `
 x )  =  C )
9 eqcom 2142 . . . . . 6  |-  ( ( (inl  |`  A ) `  x )  =  C  <-> 
C  =  ( (inl  |`  A ) `  x
) )
109rexbii 2445 . . . . 5  |-  ( E. x  e.  A  ( (inl  |`  A ) `  x )  =  C  <->  E. x  e.  A  C  =  ( (inl  |`  A ) `  x
) )
118, 10bitri 183 . . . 4  |-  ( C  e.  ran  (inl  |`  A )  <->  E. x  e.  A  C  =  ( (inl  |`  A ) `  x
) )
12 djurf1or 6949 . . . . . 6  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
13 f1ofn 5375 . . . . . 6  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  (inr  |`  B )  Fn  B )
14 fvelrnb 5476 . . . . . 6  |-  ( (inr  |`  B )  Fn  B  ->  ( C  e.  ran  (inr  |`  B )  <->  E. x  e.  B  ( (inr  |`  B ) `  x
)  =  C ) )
1512, 13, 14mp2b 8 . . . . 5  |-  ( C  e.  ran  (inr  |`  B )  <->  E. x  e.  B  ( (inr  |`  B ) `
 x )  =  C )
16 eqcom 2142 . . . . . 6  |-  ( ( (inr  |`  B ) `  x )  =  C  <-> 
C  =  ( (inr  |`  B ) `  x
) )
1716rexbii 2445 . . . . 5  |-  ( E. x  e.  B  ( (inr  |`  B ) `  x )  =  C  <->  E. x  e.  B  C  =  ( (inr  |`  B ) `  x
) )
1815, 17bitri 183 . . . 4  |-  ( C  e.  ran  (inr  |`  B )  <->  E. x  e.  B  C  =  ( (inr  |`  B ) `  x
) )
1911, 18orbi12i 754 . . 3  |-  ( ( C  e.  ran  (inl  |`  A )  \/  C  e.  ran  (inr  |`  B ) )  <->  ( E. x  e.  A  C  =  ( (inl  |`  A ) `
 x )  \/ 
E. x  e.  B  C  =  ( (inr  |`  B ) `  x
) ) )
204, 19bitri 183 . 2  |-  ( C  e.  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  <-> 
( E. x  e.  A  C  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `
 x ) ) )
213, 20bitri 183 1  |-  ( C  e.  ( A B )  <-> 
( E. x  e.  A  C  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `
 x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   E.wrex 2418    u. cun 3073   (/)c0 3367   {csn 3531    X. cxp 4544   ran crn 4547    |` cres 4548    Fn wfn 5125   -1-1-onto->wf1o 5129   ` cfv 5130   1oc1o 6313   ⊔ cdju 6929  inlcinl 6937  inrcinr 6938
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-13 1492  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  ax-un 4362
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-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1st 6045  df-2nd 6046  df-1o 6320  df-dju 6930  df-inl 6939  df-inr 6940
This theorem is referenced by:  djur  6961  exmidfodomrlemreseldju  7072
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