Theorem djuin 7168

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 4689	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4689	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	ineq12i 3372	. 2 |- ( (inl " A ) i^i (inr " B ) ) = ( ran (inl |` A ) i^i ran (inr |` B ) )
4		djuinr 7167	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
5	3, 4	eqtri 2226	1 |- ( (inl " A ) i^i (inr " B ) ) = (/)

Syntax hints:   = wceq 1373   i^i cin 3165   (/)c0 3460   ran crn 4677   |` cres 4678   "cima 4679  inlcinl 7149  inrcinr 7150

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-1st 6228  df-2nd 6229  df-1o 6504  df-inl 7151  df-inr 7152

This theorem is referenced by:  caseinl  7195  caseinr  7196  endjusym  7200  ctssdccl  7215  dju1p1e2  7307  endjudisj  7324  djuen  7325