Theorem dfqmap2 38951

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 38950	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-ec 8682	. . 3 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
3	2	mpteq2i 5198	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))
4	1, 3	eqtri 2787	1 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))

Syntax hints:   = wceq 1562  {csn 4584   ↦ cmpt 5183  dom cdm 5649   “ cima 5652  [cec 8678   QMap cqmap 38679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-opab 5165  df-mpt 5184  df-ec 8682  df-qmap 38950

This theorem is referenced by: (None)