ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eldju Unicode version

Theorem eldju 7041
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 7039 . . . 4  |-  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) )  =  ( A B )
21eqcomi 2174 . . 3  |-  ( A B )  =  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )
32eleq2i 2237 . 2  |-  ( C  e.  ( A B )  <-> 
C  e.  ( ran  (inl  |`  A )  u. 
ran  (inr  |`  B ) ) )
4 elun 3268 . . 3  |-  ( C  e.  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  <-> 
( C  e.  ran  (inl  |`  A )  \/  C  e.  ran  (inr  |`  B ) ) )
5 djulf1or 7029 . . . . . 6  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
6 f1ofn 5441 . . . . . 6  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  (inl  |`  A )  Fn  A )
7 fvelrnb 5542 . . . . . 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 2172 . . . . . 6  |-  ( ( (inl  |`  A ) `  x )  =  C  <-> 
C  =  ( (inl  |`  A ) `  x
) )
109rexbii 2477 . . . . 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 7030 . . . . . 6  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
13 f1ofn 5441 . . . . . 6  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  (inr  |`  B )  Fn  B )
14 fvelrnb 5542 . . . . . 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 2172 . . . . . 6  |-  ( ( (inr  |`  B ) `  x )  =  C  <-> 
C  =  ( (inr  |`  B ) `  x
) )
1716rexbii 2477 . . . . 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 759 . . 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 703    = wceq 1348    e. wcel 2141   E.wrex 2449    u. cun 3119   (/)c0 3414   {csn 3581    X. cxp 4607   ran crn 4610    |` cres 4611    Fn wfn 5191   -1-1-onto->wf1o 5195   ` cfv 5196   1oc1o 6385   ⊔ cdju 7010  inlcinl 7018  inrcinr 7019
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6116  df-2nd 6117  df-1o 6392  df-dju 7011  df-inl 7020  df-inr 7021
This theorem is referenced by:  djur  7042  exmidfodomrlemreseldju  7164
  Copyright terms: Public domain W3C validator