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Theorem djuinr 7064
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 7094 and djufun 7105) 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 7086). (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 7057 . . . 4 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 dff1o5 5472 . . . . 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 7058 . . . 4 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
6 dff1o5 5472 . . . . 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 3336 . 2 (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵))
10 1n0 6435 . . . . 5 1o ≠ ∅
1110necomi 2432 . . . 4 ∅ ≠ 1o
12 disjsn2 3657 . . . 4 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
1311, 12ax-mp 5 . . 3 ({∅} ∩ {1o}) = ∅
14 xpdisj1 5055 . . 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 3130  c0 3424  {csn 3594   × cxp 4626  ran crn 4629  cres 4630  1-1wf1 5215  1-1-ontowf1o 5217  1oc1o 6412  inlcinl 7046  inrcinr 7047
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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-inl 7048  df-inr 7049
This theorem is referenced by:  djuin  7065  casefun  7086  djudom  7094  djufun  7105
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