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Theorem disjxrn 36017
Description: Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
Assertion
Ref Expression
disjxrn ( Disj (𝑅𝑆) ↔ ( ≀ 𝑅 ∩ ≀ 𝑆) ⊆ I )

Proof of Theorem disjxrn
StepHypRef Expression
1 xrnrel 35665 . . 3 Rel (𝑅𝑆)
2 dfdisjALTV2 35987 . . 3 ( Disj (𝑅𝑆) ↔ ( ≀ (𝑅𝑆) ⊆ I ∧ Rel (𝑅𝑆)))
31, 2mpbiran2 709 . 2 ( Disj (𝑅𝑆) ↔ ≀ (𝑅𝑆) ⊆ I )
4 1cosscnvxrn 35755 . . 3 (𝑅𝑆) = ( ≀ 𝑅 ∩ ≀ 𝑆)
54sseq1i 3971 . 2 ( ≀ (𝑅𝑆) ⊆ I ↔ ( ≀ 𝑅 ∩ ≀ 𝑆) ⊆ I )
63, 5bitri 278 1 ( Disj (𝑅𝑆) ↔ ( ≀ 𝑅 ∩ ≀ 𝑆) ⊆ I )
Colors of variables: wff setvar class
Syntax hints:  wb 209  cin 3909  wss 3910   I cid 5432  ccnv 5527  Rel wrel 5533  cxrn 35492  ccoss 35493   Disj wdisjALTV 35527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-1st 7664  df-2nd 7665  df-ec 8266  df-xrn 35663  df-coss 35699  df-cnvrefrel 35805  df-disjALTV 35978
This theorem is referenced by:  disjorimxrn  36018
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