Theorem ecqmap 38619

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 8638	. 2 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {𝑦 ∣ 𝐴 QMap 𝑅𝑦})
2		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
3	2	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → (𝑥 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
4		eceq1 8675	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
5	4	eqeqan2d 38412	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑦 ∧ 𝑥 = 𝐴) → (𝑧 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
6	5	ancoms 458	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → (𝑧 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
7	3, 6	anbi12d 633	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → ((𝑥 ∈ dom 𝑅 ∧ 𝑧 = [𝑥]𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
8		dfqmap3 38618	. . . . . . . 8 ⊢ QMap 𝑅 = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑧 = [𝑥]𝑅)}
9	7, 8	brabga 5481	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝑦 ∈ V) → (𝐴 QMap 𝑅𝑦 ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
10	9	elvd 3445	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 QMap 𝑅𝑦 ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
11	10	abbidv 2801	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {𝑦 ∣ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)})
12		inab 4260	. . . . 5 ⊢ ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)}
13	11, 12	eqtr4di 2788	. . . 4 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}))
14		ax-5 1912	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅 → ∀𝑦 𝐴 ∈ dom 𝑅)
15		abv 3451	. . . . . . 7 ⊢ ({𝑦 ∣ 𝐴 ∈ dom 𝑅} = V ↔ ∀𝑦 𝐴 ∈ dom 𝑅)
16	14, 15	sylibr 234	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 ∈ dom 𝑅} = V)
17	16	ineq1d 4170	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅 → ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = (V ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}))
18		inv1 4349	. . . . . 6 ⊢ ({𝑦 ∣ 𝑦 = [𝐴]𝑅} ∩ V) = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
19	18	ineqcomi 4162	. . . . 5 ⊢ (V ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
20	17, 19	eqtrdi 2786	. . . 4 ⊢ (𝐴 ∈ dom 𝑅 → ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
21	13, 20	eqtrd 2770	. . 3 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
22		df-sn 4580	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
23	21, 22	eqtr4di 2788	. 2 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {[𝐴]𝑅})
24	1, 23	eqtrd 2770	1 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2713  Vcvv 3439   ∩ cin 3899  {csn 4579   class class class wbr 5097  dom cdm 5623  [cec 8633   QMap cqmap 38345

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-11 2163  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-qmap 38616

This theorem is referenced by:  ecqmap2  38620  qmapeldisjsim  39030