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Theorem eldju 6921
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 6919 . . . 4  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
21eqcomi 2121 . . 3  |-  ( A B )  =  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )
32eleq2i 2184 . 2  |-  ( C  e.  ( A B )  <-> 
C  e.  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) ) )
4 elun 3187 . . 3  |-  ( C  e.  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  <-> 
( C  e.  ran  (inl  |`  A )  \/  C  e.  ran  (inr  |`  B ) ) )
5 djulf1or 6909 . . . . . 6  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
6 f1ofn 5336 . . . . . 6  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  (inl  |`  A )  Fn  A )
7 fvelrnb 5437 . . . . . 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 2119 . . . . . 6  |-  ( ( (inl  |`  A ) `  x )  =  C  <-> 
C  =  ( (inl  |`  A ) `  x
) )
109rexbii 2419 . . . . 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 6910 . . . . . 6  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
13 f1ofn 5336 . . . . . 6  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  (inr  |`  B )  Fn  B )
14 fvelrnb 5437 . . . . . 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 2119 . . . . . 6  |-  ( ( (inr  |`  B ) `  x )  =  C  <-> 
C  =  ( (inr  |`  B ) `  x
) )
1716rexbii 2419 . . . . 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 738 . . 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 682    = wceq 1316    e. wcel 1465   E.wrex 2394    u. cun 3039   (/)c0 3333   {csn 3497    X. cxp 4507   ran crn 4510    |` cres 4511    Fn wfn 5088   -1-1-onto->wf1o 5092   ` cfv 5093   1oc1o 6274   ⊔ cdju 6890  inlcinl 6898  inrcinr 6899
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901
This theorem is referenced by:  djur  6922  exmidfodomrlemreseldju  7024
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