Theorem djuin 7123

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 ) i^i (inr " B ) ) = (/)

Proof of Theorem djuin

Step	Hyp	Ref	Expression
1		df-ima 4672	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4672	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	ineq12i 3358	. 2 |- ( (inl " A ) i^i (inr " B ) ) = ( ran (inl |` A ) i^i ran (inr |` B ) )
4		djuinr 7122	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
5	3, 4	eqtri 2214	1 |- ( (inl " A ) i^i (inr " B ) ) = (/)

Syntax hints:   = wceq 1364   i^i cin 3152   (/)c0 3446   ran crn 4660   |` cres 4661   "cima 4662  inlcinl 7104  inrcinr 7105

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 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

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 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-inl 7106  df-inr 7107

This theorem is referenced by:  caseinl  7150  caseinr  7151  endjusym  7155  ctssdccl  7170  dju1p1e2  7257  endjudisj  7270  djuen  7271