Theorem ecqmap 38831

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 8640	. 2 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {𝑦 ∣ 𝐴 QMap 𝑅𝑦})
2		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
3	2	adantr 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → (𝑥 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
4		eceq1 8677	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → [𝑥]𝑅 = [𝐴]𝑅)
5	4	eqeqan2d 38624	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑦 ∧ 𝑥 = 𝐴) → (𝑧 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
6	5	ancoms 460	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → (𝑧 = [𝑥]𝑅 ↔ 𝑦 = [𝐴]𝑅))
7	3, 6	anbi12d 639	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝑦) → ((𝑥 ∈ dom 𝑅 ∧ 𝑧 = [𝑥]𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
8		dfqmap3 38830	. . . . . . . 8 ⊢ QMap 𝑅 = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑧 = [𝑥]𝑅)}
9	7, 8	brabga 5479	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝑦 ∈ V) → (𝐴 QMap 𝑅𝑦 ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
10	9	elvd 3439	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 QMap 𝑅𝑦 ↔ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)))
11	10	abbidv 2807	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {𝑦 ∣ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)})
12		inab 4240	. . . . 5 ⊢ ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ (𝐴 ∈ dom 𝑅 ∧ 𝑦 = [𝐴]𝑅)}
13	11, 12	eqtr4di 2794	. . . 4 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}))
14		ax-5 1918	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅 → ∀𝑦 𝐴 ∈ dom 𝑅)
15		abv 3445	. . . . . . 7 ⊢ ({𝑦 ∣ 𝐴 ∈ dom 𝑅} = V ↔ ∀𝑦 𝐴 ∈ dom 𝑅)
16	14, 15	sylibr 236	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 ∈ dom 𝑅} = V)
17	16	ineq1d 4151	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅 → ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = (V ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}))
18		inv1 4329	. . . . . 6 ⊢ ({𝑦 ∣ 𝑦 = [𝐴]𝑅} ∩ V) = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
19	18	ineqcomi 4143	. . . . 5 ⊢ (V ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
20	17, 19	eqtrdi 2792	. . . 4 ⊢ (𝐴 ∈ dom 𝑅 → ({𝑦 ∣ 𝐴 ∈ dom 𝑅} ∩ {𝑦 ∣ 𝑦 = [𝐴]𝑅}) = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
21	13, 20	eqtrd 2776	. . 3 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {𝑦 ∣ 𝑦 = [𝐴]𝑅})
22		df-sn 4559	. . 3 ⊢ {[𝐴]𝑅} = {𝑦 ∣ 𝑦 = [𝐴]𝑅}
23	21, 22	eqtr4di 2794	. 2 ⊢ (𝐴 ∈ dom 𝑅 → {𝑦 ∣ 𝐴 QMap 𝑅𝑦} = {[𝐴]𝑅})
24	1, 23	eqtrd 2776	1 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548   ∈ wcel 2121  {cab 2719  Vcvv 3433   ∩ cin 3884  {csn 4558   class class class wbr 5075  dom cdm 5621  [cec 8635   QMap cqmap 38557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-mpt 5157  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8639  df-qmap 38828

This theorem is referenced by:  ecqmap2  38832  qmapeldisjsim  39242