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Theorem eldju 7081
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 7079 . . . 4  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
21eqcomi 2191 . . 3  |-  ( A B )  =  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )
32eleq2i 2254 . 2  |-  ( C  e.  ( A B )  <-> 
C  e.  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) ) )
4 elun 3288 . . 3  |-  ( C  e.  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  <-> 
( C  e.  ran  (inl  |`  A )  \/  C  e.  ran  (inr  |`  B ) ) )
5 djulf1or 7069 . . . . . 6  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
6 f1ofn 5474 . . . . . 6  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  (inl  |`  A )  Fn  A )
7 fvelrnb 5576 . . . . . 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 2189 . . . . . 6  |-  ( ( (inl  |`  A ) `  x )  =  C  <-> 
C  =  ( (inl  |`  A ) `  x
) )
109rexbii 2494 . . . . 5  |-  ( E. x  e.  A  ( (inl  |`  A ) `  x )  =  C  <->  E. x  e.  A  C  =  ( (inl  |`  A ) `  x
) )
118, 10bitri 184 . . . 4  |-  ( C  e.  ran  (inl  |`  A )  <->  E. x  e.  A  C  =  ( (inl  |`  A ) `  x
) )
12 djurf1or 7070 . . . . . 6  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
13 f1ofn 5474 . . . . . 6  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  (inr  |`  B )  Fn  B )
14 fvelrnb 5576 . . . . . 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 2189 . . . . . 6  |-  ( ( (inr  |`  B ) `  x )  =  C  <-> 
C  =  ( (inr  |`  B ) `  x
) )
1716rexbii 2494 . . . . 5  |-  ( E. x  e.  B  ( (inr  |`  B ) `  x )  =  C  <->  E. x  e.  B  C  =  ( (inr  |`  B ) `  x
) )
1815, 17bitri 184 . . . 4  |-  ( C  e.  ran  (inr  |`  B )  <->  E. x  e.  B  C  =  ( (inr  |`  B ) `  x
) )
1911, 18orbi12i 765 . . 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 184 . 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 184 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 105    \/ wo 709    = wceq 1363    e. wcel 2158   E.wrex 2466    u. cun 3139   (/)c0 3434   {csn 3604    X. cxp 4636   ran crn 4639    |` cres 4640    Fn wfn 5223   -1-1-onto->wf1o 5227   ` cfv 5228   1oc1o 6424   ⊔ cdju 7050  inlcinl 7058  inrcinr 7059
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6155  df-2nd 6156  df-1o 6431  df-dju 7051  df-inl 7060  df-inr 7061
This theorem is referenced by:  djur  7082  exmidfodomrlemreseldju  7213
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