Theorem rnqmap 39306

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 39298 and dfqs2 8702. (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 39298	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2	1	rneqi 5915	. 2 ⊢ ran QMap 𝑅 = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3		dfqs2 8702	. 2 ⊢ (dom 𝑅 / 𝑅) = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
4	2, 3	eqtr4i 2786	1 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)

This proof depends on syntax axioms:   = wceq 1570   ↦ cmpt 5185  dom cdm 5647  ran crn 5648  [cec 8693   / cqs 8694   QMap cqmap 39027

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-mpt 5186  df-cnv 5655  df-dm 5657  df-rn 5658  df-qs 8701  df-qmap 39298

This theorem is used by:  rnqmapeleldisjsim  39714  eldisjsim4  39790  eldisjs7  39793