Theorem detlem 39207

Description: If a relation is disjoint, then it is equivalent to the equivalent cosets of the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)

Hypothesis

Ref	Expression
detlem.1	⊢ Disj 𝑅

Assertion

Ref	Expression
detlem	⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Proof of Theorem detlem

Step	Hyp	Ref	Expression
1		disjim 39205	. 2 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
2		detlem.1	. . 3 ⊢ Disj 𝑅
3	2	a1i 11	. 2 ⊢ ( EqvRel ≀ 𝑅 → Disj 𝑅)
4	1, 3	impbii 209	1 ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Syntax hints:   ↔ wb 206   ≀ ccoss 38504   EqvRel weqvrel 38521   Disj wdisjALTV 38540

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-coss 38822  df-refrel 38913  df-cnvrefrel 38928  df-symrel 38945  df-trrel 38979  df-eqvrel 38990  df-disjALTV 39111

This theorem is referenced by:  det0  39211  detid  39217  detidres  39219  detinidres  39220  detxrnidres  39221