Theorem disjALTVxrnidres 38716

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 38713	. 2 ⊢ Disj I
2		disjimxrnres 38711	. 2 ⊢ ( Disj I → Disj (𝑅 ⋉ ( I ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))

Syntax hints:   I cid 5592   ↾ cres 5702   ⋉ cxrn 38136   Disj wdisjALTV 38171

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fo 6581  df-fv 6583  df-1st 8032  df-2nd 8033  df-ec 8767  df-xrn 38329  df-coss 38369  df-cnvrefrel 38485  df-funALTV 38640  df-disjALTV 38663

This theorem is referenced by:  eqvrel1cossxrnidres  38750  detxrnidres  38755  petxrnidres2  38780