Theorem rnqmap 39387

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 39379 and dfqs2 8724. (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 39379	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2	1	rneqi 5918	. 2 ⊢ ran QMap 𝑅 = ran (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3		dfqs2 8724	. 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 1569   ↦ cmpt 5185  dom cdm 5650  ran crn 5651  [cec 8715   / cqs 8716   QMap cqmap 39108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  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 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  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 5658  df-dm 5660  df-rn 5661  df-qs 8723  df-qmap 39379

This theorem is used by:  rnqmapeleldisjsim  39795  eldisjsim4  39871  eldisjs7  39874