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Theorem djuinr 6809
Description: The ranges of any left and right injections are disjoint. Remark: the restrictions seem not necessary, but the proof is not much longer than the proof of  |-  ( ran inl  i^i 
ran inr )  =  (/) (which is easily recovered from it, as in the proof of casefun 6830). (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 6802 . . . 4  |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
2 dff1o5 5275 . . . . 5  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  <->  ( (inl  |`  A ) : A -1-1-> ( {
(/) }  X.  A
)  /\  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) ) )
32simprbi 270 . . . 4  |-  ( (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A
)  ->  ran  (inl  |`  A )  =  ( { (/) }  X.  A ) )
41, 3ax-mp 7 . . 3  |-  ran  (inl  |`  A )  =  ( { (/) }  X.  A
)
5 djurf1or 6803 . . . 4  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
6 dff1o5 5275 . . . . 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 270 . . . 4  |-  ( (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B
)  ->  ran  (inr  |`  B )  =  ( { 1o }  X.  B ) )
85, 7ax-mp 7 . . 3  |-  ran  (inr  |`  B )  =  ( { 1o }  X.  B )
94, 8ineq12i 3200 . 2  |-  ( ran  (inl  |`  A )  i^i 
ran  (inr  |`  B ) )  =  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  B ) )
10 1n0 6211 . . . . 5  |-  1o  =/=  (/)
1110necomi 2341 . . . 4  |-  (/)  =/=  1o
12 disjsn2 3509 . . . 4  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
1311, 12ax-mp 7 . . 3  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
14 xpdisj1 4868 . . 3  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  ->  ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  B ) )  =  (/) )
1513, 14ax-mp 7 . 2  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  B ) )  =  (/)
169, 15eqtri 2109 1  |-  ( ran  (inl  |`  A )  i^i 
ran  (inr  |`  B ) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1290    =/= wne 2256    i^i cin 2999   (/)c0 3287   {csn 3450    X. cxp 4450   ran crn 4453    |` cres 4454   -1-1->wf1 5025   -1-1-onto->wf1o 5027   1oc1o 6188  inlcinl 6791  inrcinr 6792
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-1st 5925  df-2nd 5926  df-1o 6195  df-inl 6793  df-inr 6794
This theorem is referenced by:  casefun  6830  djudom  6837  djufun  6840
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