Theorem rnqmap 38657

Description: The range of the quotient map is the quotient carrier. It lets us replace quotient-carrier reasoning by map/range reasoning (and conversely) via df-qmap 38649 and dfqs2 8644. (Contributed by Peter Mazsa, 12-Feb-2026.)

Assertion

Ref	Expression
rnqmap	⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)

Proof of Theorem rnqmap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-qmap 38649	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2	1	rneqi 5887	. 2 ⊢ ran QMap 𝑅 = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3		dfqs2 8644	. 2 ⊢ (dom 𝑅 / 𝑅) = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
4	2, 3	eqtr4i 2763	1 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)

Syntax hints:   = wceq 1542   ↦ cmpt 5180  dom cdm 5625  ran crn 5626  [cec 8635   / cqs 8636   QMap cqmap 38378

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 5242  ax-nul 5252  ax-pr 5378

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-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-mpt 5181  df-cnv 5633  df-dm 5635  df-rn 5636  df-qs 8643  df-qmap 38649

This theorem is referenced by:  rnqmapeleldisjsim  39065  eldisjsim4  39141  eldisjs7  39144