Theorem qmapeldisjsim 39712

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 39677	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
2		disjimeceqim2 39657	. . . 4 ⊢ ( Disj QMap 𝑅 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)))
3		dmqmap 39305	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)
4	3	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∈ dom QMap 𝑅 ↔ 𝐴 ∈ dom 𝑅))
5	3	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐵 ∈ dom QMap 𝑅 ↔ 𝐵 ∈ dom 𝑅))
6	4, 5	anbi12d 644	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
7	6	pm5.32i 585	. . . . . . . 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 39301	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
10		ecqmap 39301	. . . . . . . . . . 11 ⊢ (𝐵 ∈ dom 𝑅 → [𝐵] QMap 𝑅 = {[𝐵]𝑅})
11	9, 10	eqeqan12d 2774	. . . . . . . . . 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 8699	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → [𝐴]𝑅 ∈ V)
14		sneqbg 4802	. . . . . . . . . . 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 520	. . . . . . . 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 456	. . . . . 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 456	. . . . . 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 1128	1 ⊢ ((𝑅 ∈ 𝑉 ∧ QMap 𝑅 ∈ Disjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450  {csn 4583  dom cdm 5647  [cec 8693   QMap cqmap 39027   Disjs cdisjs 39070   Disj wdisjALTV 39071

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ec 8697  df-rels 39292  df-qmap 39298  df-coss 39353  df-ssr 39430  df-cnvrefs 39457  df-cnvrefrels 39458  df-cnvrefrel 39459  df-disjss 39640  df-disjs 39641  df-disjALTV 39642

This theorem is used by:  qmapeldisjsbi  39713  eldisjs6  39792