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

Theorem djuinr 7096
Description: The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 7126 and djufun 7137) while the simpler statement  |-  ( ran inl  i^i 
ran inr )  =  (/) is easily recovered from it by substituting  _V for both  A and  B as done in casefun 7118). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djuinr  |-  ( ran  (inl  |`  A )  i^i 
ran  (inr  |`  B ) )  =  (/)

Proof of Theorem djuinr
StepHypRef Expression
1 djulf1or 7089 . . . 4  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
2 dff1o5 5492 . . . . 5  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  <->  ( (inl  |`  A ) : A -1-1-> ( {
(/) }  X.  A
)  /\  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) ) )
32simprbi 275 . . . 4  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) )
41, 3ax-mp 5 . . 3  |-  ran  (inl  |`  A )  =  ( { (/) }  X.  A
)
5 djurf1or 7090 . . . 4  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
6 dff1o5 5492 . . . . 5  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  <->  ( (inr  |`  B ) : B -1-1-> ( { 1o }  X.  B
)  /\  ran  (inr  |`  B )  =  ( { 1o }  X.  B ) ) )
76simprbi 275 . . . 4  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  ran  (inr  |`  B )  =  ( { 1o }  X.  B ) )
85, 7ax-mp 5 . . 3  |-  ran  (inr  |`  B )  =  ( { 1o }  X.  B )
94, 8ineq12i 3349 . 2  |-  ( ran  (inl  |`  A )  i^i 
ran  (inr  |`  B ) )  =  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  B ) )
10 1n0 6461 . . . . 5  |-  1o  =/=  (/)
1110necomi 2445 . . . 4  |-  (/)  =/=  1o
12 disjsn2 3673 . . . 4  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
1311, 12ax-mp 5 . . 3  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
14 xpdisj1 5074 . . 3  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  ->  ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  B ) )  =  (/) )
1513, 14ax-mp 5 . 2  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  B ) )  =  (/)
169, 15eqtri 2210 1  |-  ( ran  (inl  |`  A )  i^i 
ran  (inr  |`  B ) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1364    =/= wne 2360    i^i cin 3143   (/)c0 3437   {csn 3610    X. cxp 4645   ran crn 4648    |` cres 4649   -1-1->wf1 5235   -1-1-onto->wf1o 5237   1oc1o 6438  inlcinl 7078  inrcinr 7079
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-1st 6169  df-2nd 6170  df-1o 6445  df-inl 7080  df-inr 7081
This theorem is referenced by:  djuin  7097  casefun  7118  djudom  7126  djufun  7137
  Copyright terms: Public domain W3C validator