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Theorem partim 37270
Description: Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 37269. (Contributed by Peter Mazsa, 17-Sep-2021.)
Assertion
Ref Expression
partim (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)

Proof of Theorem partim
StepHypRef Expression
1 partim2 37269 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))
2 dfpart2 37231 . 2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
3 dferALTV2 37130 . 2 ( ≀ 𝑅 ErALTV 𝐴 ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))
41, 2, 33imtr4i 291 1 (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  dom cdm 5633   / cqs 8647  ccoss 36634   EqvRel weqvrel 36651   ErALTV werALTV 36660   Disj wdisjALTV 36668   Part wpart 36673
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ec 8650  df-qs 8654  df-coss 36873  df-refrel 36974  df-cnvrefrel 36989  df-symrel 37006  df-trrel 37036  df-eqvrel 37047  df-dmqs 37101  df-erALTV 37126  df-disjALTV 37167  df-part 37228
This theorem is referenced by:  partimeq  37271  partimcomember  37297
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