Theorem djuin 6864

Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)

Assertion

Ref	Expression
djuin	⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Proof of Theorem djuin

Step	Hyp	Ref	Expression
1		df-ima 4490	. . 3 ⊢ (inl “ 𝐴) = ran (inl ↾ 𝐴)
2		df-ima 4490	. . 3 ⊢ (inr “ 𝐵) = ran (inr ↾ 𝐵)
3	1, 2	ineq12i 3222	. 2 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵))
4		djuinr 6863	. 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅
5	3, 4	eqtri 2120	1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Syntax hints:   = wceq 1299   ∩ cin 3020  ∅c0 3310  ran crn 4478   ↾ cres 4479   “ cima 4480  inlcinl 6845  inrcinr 6846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1st 5969  df-2nd 5970  df-1o 6243  df-inl 6847  df-inr 6848

This theorem is referenced by:  caseinl  6891  caseinr  6892  endjusym  6896  ctssdclemr  6911  dju1p1e2  6962  endjudisj  6970  djuen  6971