Theorem ecqmap2 38771

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 38770	. 2 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
2		snecg 8724	. 2 ⊢ (𝐴 ∈ dom 𝑅 → {[𝐴]𝑅} = ({𝐴} / 𝑅))
3	1, 2	eqtrd 2771	1 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = ({𝐴} / 𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4567  dom cdm 5631  [cec 8641   / cqs 8642   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-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  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-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8645  df-qs 8649  df-qmap 38767

This theorem is referenced by: (None)