Theorem dfqmap2 38650

Description: Alternate definition of the quotient map: QMap in image-of-singleton form. (Contributed by Peter Mazsa, 14-Feb-2026.)

Assertion

Ref	Expression
dfqmap2	⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))

Distinct variable group:   𝑥,𝑅

Proof of Theorem dfqmap2

Step	Hyp	Ref	Expression
1		df-qmap 38649	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-ec 8639	. . 3 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
3	2	mpteq2i 5195	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))
4	1, 3	eqtri 2760	1 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))

Syntax hints:   = wceq 1542  {csn 4581   ↦ cmpt 5180  dom cdm 5625   “ cima 5628  [cec 8635   QMap cqmap 38378

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-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-opab 5162  df-mpt 5181  df-ec 8639  df-qmap 38649

This theorem is referenced by: (None)