Theorem ecqmap 39572

Description: QMap fibers are singletons of blocks. Makes QMap behave like a "block constructor function" on dom 𝑅. (Contributed by Peter Mazsa, 14-Feb-2026.)

Assertion

Ref	Expression
ecqmap	⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})

Proof of Theorem ecqmap

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfec2 8745	. 2 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {𝑦 ∣ 𝐴 QMap 𝑅𝑦})
2		eleq1 2852	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
3	2	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → (𝑥 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
4		eceq1 8782	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
5	4	eqeqan2d 39365	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑦 ∧ 𝑥 = 𝐴) → (𝑧 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
6	5	ancoms 464	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → (𝑧 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
7	3, 6	anbi12d 644	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → ((𝑥 ∈ dom 𝑅 ∧ 𝑧 = [𝑥]𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
8		dfqmap3 39571	. . . . . . . 8 ⊢ QMap 𝑅 = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑧 = [𝑥]𝑅)}
9	7, 8	brabga 5512	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝑦 ∈ V) → (𝐴 QMap 𝑅𝑦 ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
10	9	elvd 3460	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 QMap 𝑅𝑦 ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
11	10	abbidv 2829	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {𝑦 ∣ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)})
12		inab 4257	. . . . 5 ⊢ ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)}
13	11, 12	eqtr4di 2816	. . . 4 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}))
14		ax-5 1945	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅 → ∀𝑦 𝐴 ∈ dom 𝑅)
15		abv 3466	. . . . . . 7 ⊢ ({𝑦 ∣ 𝐴 ∈ dom 𝑅} = V ↔ ∀𝑦 𝐴 ∈ dom 𝑅)
16	14, 15	sylibr 237	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 ∈ dom 𝑅} = V)
17	16	ineq1d 4119	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅 → ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = (V ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}))
18		inv1 4351	. . . . . 6 ⊢ ({𝑦 ∣ 𝑦 = [𝐴]𝑅} ∩ V) = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
19	18	ineqcomi 4111	. . . . 5 ⊢ (V ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
20	17, 19	eqtrdi 2814	. . . 4 ⊢ (𝐴 ∈ dom 𝑅 → ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
21	13, 20	eqtrd 2798	. . 3 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
22		df-sn 4589	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
23	21, 22	eqtr4di 2816	. 2 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {[𝐴]𝑅})
24	1, 23	eqtrd 2798	1 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   ∩ cin 4092  {csn 4588   class class class wbr 5106  dom cdm 5655  [cec 8740   QMap cqmap 39298

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-qmap 39569

This theorem is used by:  ecqmap2  39573  qmapeldisjsim  39983