Theorem rnqmap 38795

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 38787 and dfqs2 8645. (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 38787	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2	1	rneqi 5888	. 2 ⊢ ran QMap 𝑅 = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3		dfqs2 8645	. 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 5167  dom cdm 5626  ran crn 5627  [cec 8636   / cqs 8637   QMap cqmap 38516

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 5232  ax-pr 5372

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 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-cnv 5634  df-dm 5636  df-rn 5637  df-qs 8644  df-qmap 38787

This theorem is referenced by:  rnqmapeleldisjsim  39203  eldisjsim4  39279  eldisjs7  39282