Theorem dmqmap 38774

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 38767	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		ecexg 8647	. . 3 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ dom 𝑅) → [𝑥]𝑅 ∈ V)
4	1, 3	dmmptd 6643	1 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  dom cdm 5631  [cec 8641   QMap cqmap 38496

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 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8645  df-qmap 38767

This theorem is referenced by:  qmapeldisjsim  39181