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Theorem eldju 6868
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 6866 . . . 4  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
21eqcomi 2104 . . 3  |-  ( A B )  =  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )
32eleq2i 2166 . 2  |-  ( C  e.  ( A B )  <-> 
C  e.  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) ) )
4 elun 3164 . . 3  |-  ( C  e.  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  <-> 
( C  e.  ran  (inl  |`  A )  \/  C  e.  ran  (inr  |`  B ) ) )
5 djulf1or 6856 . . . . . 6  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
6 f1ofn 5302 . . . . . 6  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  (inl  |`  A )  Fn  A )
7 fvelrnb 5401 . . . . . 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 2102 . . . . . 6  |-  ( ( (inl  |`  A ) `  x )  =  C  <-> 
C  =  ( (inl  |`  A ) `  x
) )
109rexbii 2401 . . . . 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 6857 . . . . . 6  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
13 f1ofn 5302 . . . . . 6  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  (inr  |`  B )  Fn  B )
14 fvelrnb 5401 . . . . . 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 2102 . . . . . 6  |-  ( ( (inr  |`  B ) `  x )  =  C  <-> 
C  =  ( (inr  |`  B ) `  x
) )
1716rexbii 2401 . . . . 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 722 . . 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 670    = wceq 1299    e. wcel 1448   E.wrex 2376    u. cun 3019   (/)c0 3310   {csn 3474    X. cxp 4475   ran crn 4478    |` cres 4479    Fn wfn 5054   -1-1-onto->wf1o 5058   ` cfv 5059   1oc1o 6236   ⊔ cdju 6837  inlcinl 6845  inrcinr 6846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1st 5969  df-2nd 5970  df-1o 6243  df-dju 6838  df-inl 6847  df-inr 6848
This theorem is referenced by:  djur  6869  exmidfodomrlemreseldju  6965
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