Theorem dmqmap 39576

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 39569	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		ecexg 8746	. . 3 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V)
3	2	adantr 486	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ dom 𝑅) → [𝑥]𝑅 ∈ V)
4	1, 3	dmmptd 6703	1 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  dom cdm 5655  [cec 8740   QMap cqmap 39298

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7772

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-qmap 39569

This theorem is used by:  qmapeldisjsim  39983