Theorem dfqmap2 39124

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 39123	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-ec 8694	. . 3 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
3	2	mpteq2i 5206	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))
4	1, 3	eqtri 2785	1 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))

This proof depends on syntax axioms:   = wceq 1569  {csn 4588   ↦ cmpt 5191  dom cdm 5660   “ cima 5663  [cec 8690   QMap cqmap 38852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-opab 5173  df-mpt 5192  df-ec 8694  df-qmap 39123

This theorem is used by: (None)