Theorem dfqmap2 39198

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 39197	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-ec 8701	. . 3 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥})
3	2	mpteq2i 5201	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))
4	1, 3	eqtri 2783	1 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))

This proof depends on syntax axioms:   = wceq 1570  {csn 4584   ↦ cmpt 5186  dom cdm 5655   “ cima 5658  [cec 8697   QMap cqmap 38926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-opab 5168  df-mpt 5187  df-ec 8701  df-qmap 39197

This theorem is used by: (None)