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Theorem djuin 7037
Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
Assertion
Ref Expression
djuin  |-  ( (inl " A )  i^i  (inr " B ) )  =  (/)

Proof of Theorem djuin
StepHypRef Expression
1 df-ima 4622 . . 3  |-  (inl " A )  =  ran  (inl  |`  A )
2 df-ima 4622 . . 3  |-  (inr " B )  =  ran  (inr  |`  B )
31, 2ineq12i 3326 . 2  |-  ( (inl " A )  i^i  (inr " B ) )  =  ( ran  (inl  |`  A )  i^i  ran  (inr  |`  B ) )
4 djuinr 7036 . 2  |-  ( ran  (inl  |`  A )  i^i 
ran  (inr  |`  B ) )  =  (/)
53, 4eqtri 2191 1  |-  ( (inl " A )  i^i  (inr " B ) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1348    i^i cin 3120   (/)c0 3414   ran crn 4610    |` cres 4611   "cima 4612  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-fal 1354  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-ne 2341  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-ima 4622  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-inl 7020  df-inr 7021
This theorem is referenced by:  caseinl  7064  caseinr  7065  endjusym  7069  ctssdccl  7084  dju1p1e2  7161  endjudisj  7174  djuen  7175
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