Theorem qmapeldisjsim 39459

Description: Injectivity of coset map from QMap being disjoint (implication form): under the Disjs condition on QMap 𝑅, the coset assignment is injective on dom 𝑅. (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
qmapeldisjsim	⊢ ((𝑅 ∈ 𝑉 ∧ QMap 𝑅 ∈ Disjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))

Proof of Theorem qmapeldisjsim

Step	Hyp	Ref	Expression
1		qmapeldisjs 39424	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
2		disjimeceqim2 39404	. . . 4 ⊢ ( Disj QMap 𝑅 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)))
3		dmqmap 39052	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)
4	3	eleq2d 2856	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∈ dom QMap 𝑅 ↔ 𝐴 ∈ dom 𝑅))
5	3	eleq2d 2856	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐵 ∈ dom QMap 𝑅 ↔ 𝐵 ∈ dom 𝑅))
6	4, 5	anbi12d 643	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
7	6	pm5.32i 584	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
8	7	imbi1i 352	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅)) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)))
9		ecqmap 39048	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
10		ecqmap 39048	. . . . . . . . . . 11 ⊢ (𝐵 ∈ dom 𝑅 → [𝐵] QMap 𝑅 = {[𝐵]𝑅})
11	9, 10	eqeqan12d 2784	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 ↔ {[𝐴]𝑅} = {[𝐵]𝑅}))
12	11	imbi1d 344	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵) ↔ ({[𝐴]𝑅} = {[𝐵]𝑅} → 𝐴 = 𝐵)))
13		ecexg 8701	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → [𝐴]𝑅 ∈ V)
14		sneqbg 4813	. . . . . . . . . . 11 ⊢ ([𝐴]𝑅 ∈ V → ({[𝐴]𝑅} = {[𝐵]𝑅} ↔ [𝐴]𝑅 = [𝐵]𝑅))
15	13, 14	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ({[𝐴]𝑅} = {[𝐵]𝑅} ↔ [𝐴]𝑅 = [𝐵]𝑅))
16	15	imbi1d 344	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (({[𝐴]𝑅} = {[𝐵]𝑅} → 𝐴 = 𝐵) ↔ ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
17	12, 16	sylan9bbr 519	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → (([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵) ↔ ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
18	17	pm5.74i 274	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
19	8, 18	bitri 278	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅)) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)) ↔ ((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
20		impexp 455	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅)) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)) ↔ (𝑅 ∈ 𝑉 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵))))
21		impexp 455	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)) ↔ (𝑅 ∈ 𝑉 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
22	19, 20, 21	3bitr3i 304	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵))) ↔ (𝑅 ∈ 𝑉 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
23	22	pm5.74ri 275	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)) ↔ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
24	2, 23	imbitrid 247	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
25	1, 24	sylbid 243	. 2 ⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
26	25	3imp 1126	1 ⊢ ((𝑅 ∈ 𝑉 ∧ QMap 𝑅 ∈ Disjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  Vcvv 3462  {csn 4594  dom cdm 5665  [cec 8695   QMap cqmap 38774   Disjs cdisjs 38817   Disj wdisjALTV 38818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ec 8699  df-rels 39039  df-qmap 39045  df-coss 39100  df-ssr 39177  df-cnvrefs 39204  df-cnvrefrels 39205  df-cnvrefrel 39206  df-disjss 39387  df-disjs 39388  df-disjALTV 39389

This theorem is referenced by:  qmapeldisjsbi  39460  eldisjs6  39539