Theorem disjALTVxrnidres 38299

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 38296	. 2 ⊢ Disj I
2		disjimxrnres 38294	. 2 ⊢ ( Disj I → Disj (𝑅 ⋉ ( I ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))

Syntax hints:   I cid 5574   ↾ cres 5679   ⋉ cxrn 37717   Disj wdisjALTV 37752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428  ax-un 7739

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-fo 6553  df-fv 6555  df-1st 7992  df-2nd 7993  df-ec 8725  df-xrn 37912  df-coss 37952  df-cnvrefrel 38068  df-funALTV 38223  df-disjALTV 38246

This theorem is referenced by:  eqvrel1cossxrnidres  38333  detxrnidres  38338  petxrnidres2  38363