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Theorem djuinr 7061
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 7091 and djufun 7102) while the simpler statement (ran inl ∩ ran inr) = ∅ is easily recovered from it by substituting V for both 𝐴 and 𝐵 as done in casefun 7083). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djuinr (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅

Proof of Theorem djuinr
StepHypRef Expression
1 djulf1or 7054 . . . 4 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 dff1o5 5470 . . . . 5 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) ↔ ((inl ↾ 𝐴):𝐴1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴)))
32simprbi 275 . . . 4 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
41, 3ax-mp 5 . . 3 ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5 djurf1or 7055 . . . 4 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
6 dff1o5 5470 . . . . 5 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) ↔ ((inr ↾ 𝐵):𝐵1-1→({1o} × 𝐵) ∧ ran (inr ↾ 𝐵) = ({1o} × 𝐵)))
76simprbi 275 . . . 4 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵))
85, 7ax-mp 5 . . 3 ran (inr ↾ 𝐵) = ({1o} × 𝐵)
94, 8ineq12i 3334 . 2 (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵))
10 1n0 6432 . . . . 5 1o ≠ ∅
1110necomi 2432 . . . 4 ∅ ≠ 1o
12 disjsn2 3655 . . . 4 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
1311, 12ax-mp 5 . . 3 ({∅} ∩ {1o}) = ∅
14 xpdisj1 5053 . . 3 (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅)
1513, 14ax-mp 5 . 2 (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅
169, 15eqtri 2198 1 (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wne 2347  cin 3128  c0 3422  {csn 3592   × cxp 4624  ran crn 4627  cres 4628  1-1wf1 5213  1-1-ontowf1o 5215  1oc1o 6409  inlcinl 7043  inrcinr 7044
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-inl 7045  df-inr 7046
This theorem is referenced by:  djuin  7062  casefun  7083  djudom  7091  djufun  7102
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