Theorem djuinr 7403

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 7433 and djufun 7444) 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 7425). (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 7396	. . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
2		dff1o5 5648	. . . . 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 7397	. . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
6		dff1o5 5648	. . . . 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 3430	. 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵))
10		1n0 6705	. . . . 5 ⊢ 1o ≠ ∅
11	10	necomi 2505	. . . 4 ⊢ ∅ ≠ 1o
12		disjsn2 3772	. . . 4 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
13	11, 12	ax-mp 5	. . 3 ⊢ ({∅} ∩ {1o}) = ∅
14		xpdisj1 5212	. . 3 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅)
15	13, 14	ax-mp 5	. 2 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅
16	9, 15	eqtri 2259	1 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅

This proof depends on syntax axioms:   = wceq 1402   ≠ wne 2420   ∩ cin 3219  ∅c0 3520  {csn 3709   × cxp 4772  ran crn 4775   ↾ cres 4776  –1-1→wf1 5374  –1-1-onto→wf1o 5376  1_oc1o 6680  inlcinl 7385  inrcinr 7386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-inl 7387  df-inr 7388

This theorem is used by:  djuin  7404  casefun  7425  djudom  7433  djufun  7444