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Theorem djur 6920
Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
Assertion
Ref Expression
djur  |-  ( C  e.  ( A B )  <-> 
( E. x  e.  A  C  =  (inl
`  x )  \/ 
E. x  e.  B  C  =  (inr `  x
) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem djur
StepHypRef Expression
1 eldju 6919 . 2  |-  ( C  e.  ( A B )  <-> 
( E. x  e.  A  C  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `
 x ) ) )
2 fvres 5411 . . . . 5  |-  ( x  e.  A  ->  (
(inl  |`  A ) `  x )  =  (inl
`  x ) )
32eqeq2d 2127 . . . 4  |-  ( x  e.  A  ->  ( C  =  ( (inl  |`  A ) `  x
)  <->  C  =  (inl `  x ) ) )
43rexbiia 2425 . . 3  |-  ( E. x  e.  A  C  =  ( (inl  |`  A ) `
 x )  <->  E. x  e.  A  C  =  (inl `  x ) )
5 fvres 5411 . . . . 5  |-  ( x  e.  B  ->  (
(inr  |`  B ) `  x )  =  (inr
`  x ) )
65eqeq2d 2127 . . . 4  |-  ( x  e.  B  ->  ( C  =  ( (inr  |`  B ) `  x
)  <->  C  =  (inr `  x ) ) )
76rexbiia 2425 . . 3  |-  ( E. x  e.  B  C  =  ( (inr  |`  B ) `
 x )  <->  E. x  e.  B  C  =  (inr `  x ) )
84, 7orbi12i 736 . 2  |-  ( ( E. x  e.  A  C  =  ( (inl  |`  A ) `  x
)  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `
 x ) )  <-> 
( E. x  e.  A  C  =  (inl
`  x )  \/ 
E. x  e.  B  C  =  (inr `  x
) ) )
91, 8bitri 183 1  |-  ( C  e.  ( A B )  <-> 
( E. x  e.  A  C  =  (inl
`  x )  \/ 
E. x  e.  B  C  =  (inr `  x
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   E.wrex 2392    |` cres 4509   ` cfv 5091   ⊔ cdju 6888  inlcinl 6896  inrcinr 6897
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  djuss  6921  updjud  6933  omp1eomlem  6945  0ct  6958  ctmlemr  6959  ctssdclemn0  6961  fodjuomnilemdc  6982  exmidfodomrlemeldju  7019
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