Theorem detlem 38887

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 38885	. 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 38228   EqvRel weqvrel 38245   Disj wdisjALTV 38262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-coss 38519  df-refrel 38610  df-cnvrefrel 38625  df-symrel 38642  df-trrel 38676  df-eqvrel 38687  df-disjALTV 38809

This theorem is referenced by:  det0  38891  detid  38897  detidres  38899  detinidres  38900  detxrnidres  38901