Theorem dmqmap 38623

Description: QMap preserves the domain. Confirms that QMap is defined exactly on the points where cosets [𝑥]𝑅 make sense (those in dom 𝑅). (Contributed by Peter Mazsa, 14-Feb-2026.)

Assertion

Ref	Expression
dmqmap	⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)

Proof of Theorem dmqmap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-qmap 38616	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		ecexg 8639	. . 3 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ dom 𝑅) → [𝑥]𝑅 ∈ V)
4	1, 3	dmmptd 6636	1 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3439  dom cdm 5623  [cec 8633   QMap cqmap 38345

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-qmap 38616

This theorem is referenced by:  qmapeldisjsim  39030