Theorem detlem 39349

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 39347	. 2 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)
2		detlem.1	. . 3 ⊢ Disj 𝑅
3	2	a1i 11	. 2 ⊢ ( EqvRel ≀ 𝑅 → Disj 𝑅)
4	1, 3	impbii 211	1 ⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Syntax hints:   ↔ wb 208   ≀ ccoss 38646   EqvRel weqvrel 38663   Disj wdisjALTV 38682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-coss 38964  df-refrel 39055  df-cnvrefrel 39070  df-symrel 39087  df-trrel 39121  df-eqvrel 39132  df-disjALTV 39253

This theorem is referenced by:  det0  39353  detid  39359  detidres  39361  detinidres  39362  detxrnidres  39363