Theorem unidmqs 38636

Description: The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)

Assertion

Ref	Expression
unidmqs	⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅))

Proof of Theorem unidmqs

Step	Hyp	Ref	Expression
1		resexg 6047	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2		rnresequniqs 38314	. . . 4 ⊢ ((𝑅 ↾ dom 𝑅) ∈ V → ran (𝑅 ↾ dom 𝑅) = ∪ (dom 𝑅 / 𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ↾ dom 𝑅) = ∪ (dom 𝑅 / 𝑅))
4		resdm 6046	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
5	4	rneqd 5952	. . 3 ⊢ (Rel 𝑅 → ran (𝑅 ↾ dom 𝑅) = ran 𝑅)
6	3, 5	sylan9req 2796	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)
7	6	ex 412	1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∪ cuni 4912  dom cdm 5689  ran crn 5690   ↾ cres 5691  Rel wrel 5694   / cqs 8743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750

This theorem is referenced by:  unidmqseq  38637