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Theorem djuinr 6958
 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 6988 and djufun 6999) 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 6980). (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 6951 . . . 4 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 dff1o5 5385 . . . . 5 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) ↔ ((inl ↾ 𝐴):𝐴1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴)))
32simprbi 273 . . . 4 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
41, 3ax-mp 5 . . 3 ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5 djurf1or 6952 . . . 4 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
6 dff1o5 5385 . . . . 5 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) ↔ ((inr ↾ 𝐵):𝐵1-1→({1o} × 𝐵) ∧ ran (inr ↾ 𝐵) = ({1o} × 𝐵)))
76simprbi 273 . . . 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 3281 . 2 (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵))
10 1n0 6338 . . . . 5 1o ≠ ∅
1110necomi 2394 . . . 4 ∅ ≠ 1o
12 disjsn2 3595 . . . 4 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
1311, 12ax-mp 5 . . 3 ({∅} ∩ {1o}) = ∅
14 xpdisj1 4972 . . 3 (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅)
1513, 14ax-mp 5 . 2 (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅
169, 15eqtri 2161 1 (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ≠ wne 2309   ∩ cin 3076  ∅c0 3369  {csn 3533   × cxp 4546  ran crn 4549   ↾ cres 4550  –1-1→wf1 5129  –1-1-onto→wf1o 5131  1oc1o 6315  inlcinl 6940  inrcinr 6941 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-1st 6047  df-2nd 6048  df-1o 6322  df-inl 6942  df-inr 6943 This theorem is referenced by:  djuin  6959  casefun  6980  djudom  6988  djufun  6999
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