Theorem ecqmap2 38620

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 38619	. 2 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
2		snecg 8716	. 2 ⊢ (𝐴 ∈ dom 𝑅 → {[𝐴]𝑅} = ({𝐴} / 𝑅))
3	1, 2	eqtrd 2770	1 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = ({𝐴} / 𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4579  dom cdm 5623  [cec 8633   / cqs 8634   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-11 2163  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  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-br 5098  df-opab 5160  df-mpt 5179  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-qs 8641  df-qmap 38616

This theorem is referenced by: (None)