Theorem disjALTVxrnidres 36793

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 36790	. 2 ⊢ Disj I
2		disjimxrnres 36788	. 2 ⊢ ( Disj I → Disj (𝑅 ⋉ ( I ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))

Syntax hints:   I cid 5479   ↾ cres 5582   ⋉ cxrn 36259   Disj wdisjALTV 36294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804  df-2nd 7805  df-ec 8458  df-xrn 36428  df-coss 36464  df-cnvrefrel 36570  df-funALTV 36720  df-disjALTV 36743

This theorem is referenced by: (None)