Theorem rnqmap 39131

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 39123 and dfqs2 8699. (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 39123	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2	1	rneqi 5926	. 2 ⊢ ran QMap 𝑅 = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3		dfqs2 8699	. 2 ⊢ (dom 𝑅 / 𝑅) = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
4	2, 3	eqtr4i 2788	1 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)

This proof depends on syntax axioms:   = wceq 1569   ↦ cmpt 5191  dom cdm 5660  ran crn 5661  [cec 8690   / cqs 8691   QMap cqmap 38852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-mpt 5192  df-cnv 5668  df-dm 5670  df-rn 5671  df-qs 8698  df-qmap 39123

This theorem is used by:  rnqmapeleldisjsim  39539  eldisjsim4  39615  eldisjs7  39618