Theorem ecqmap2 39302

Description: Fiber of QMap equals singleton quotient: a conceptual bridge between "map fibers" and quotients. (Contributed by Peter Mazsa, 19-Feb-2026.)

Assertion

Ref	Expression
ecqmap2	⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = ({𝐴} / 𝑅))

Proof of Theorem ecqmap2

Step	Hyp	Ref	Expression
1		ecqmap 39301	. 2 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
2		snecg 8776	. 2 ⊢ (𝐴 ∈ dom 𝑅 → {[𝐴]𝑅} = ({𝐴} / 𝑅))
3	1, 2	eqtrd 2795	1 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = ({𝐴} / 𝑅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {csn 4583  dom cdm 5647  [cec 8693   / cqs 8694   QMap cqmap 39027

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  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 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  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-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8697  df-qs 8701  df-qmap 39298

This theorem is used by: (None)