Theorem djuinr 7124

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 7154 and djufun 7165) 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 7146). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)

Assertion

Ref	Expression
djuinr	⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅

Proof of Theorem djuinr

Step	Hyp	Ref	Expression
1		djulf1or 7117	. . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
2		dff1o5 5510	. . . . 5 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) ↔ ((inl ↾ 𝐴):𝐴–1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴)))
3	2	simprbi 275	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴))
4	1, 3	ax-mp 5	. . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴)
5		djurf1or 7118	. . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
6		dff1o5 5510	. . . . 5 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) ↔ ((inr ↾ 𝐵):𝐵–1-1→({1o} × 𝐵) ∧ ran (inr ↾ 𝐵) = ({1o} × 𝐵)))
7	6	simprbi 275	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵))
8	5, 7	ax-mp 5	. . 3 ⊢ ran (inr ↾ 𝐵) = ({1o} × 𝐵)
9	4, 8	ineq12i 3359	. 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵))
10		1n0 6487	. . . . 5 ⊢ 1o ≠ ∅
11	10	necomi 2449	. . . 4 ⊢ ∅ ≠ 1o
12		disjsn2 3682	. . . 4 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
13	11, 12	ax-mp 5	. . 3 ⊢ ({∅} ∩ {1o}) = ∅
14		xpdisj1 5091	. . 3 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅
16	9, 15	eqtri 2214	1 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅

Syntax hints:   = wceq 1364   ≠ wne 2364   ∩ cin 3153  ∅c0 3447  {csn 3619   × cxp 4658  ran crn 4661   ↾ cres 4662  –1-1→wf1 5252  –1-1-onto→wf1o 5254  1_oc1o 6464  inlcinl 7106  inrcinr 7107

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-inl 7108  df-inr 7109

This theorem is referenced by:  djuin  7125  casefun  7146  djudom  7154  djufun  7165