Theorem disjALTVxrnidres 38752

Description: The class of range Cartesian product with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 25-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.)

Assertion

Ref	Expression
disjALTVxrnidres	⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))

Proof of Theorem disjALTVxrnidres

Step	Hyp	Ref	Expression
1		disjALTVid 38749	. 2 ⊢ Disj I
2		disjimxrnres 38747	. 2 ⊢ ( Disj I → Disj (𝑅 ⋉ ( I ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))

Syntax hints:   I cid 5583   ↾ cres 5692   ⋉ cxrn 38173   Disj wdisjALTV 38208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-ima 5703  df-iota 6519  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-1st 8019  df-2nd 8020  df-ec 8752  df-xrn 38365  df-coss 38405  df-cnvrefrel 38521  df-funALTV 38676  df-disjALTV 38699

This theorem is referenced by:  eqvrel1cossxrnidres  38786  detxrnidres  38791  petxrnidres2  38816