Theorem qmapeldisjsim 39140

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 39105	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
2		disjimeceqim2 39085	. . . 4 ⊢ ( Disj QMap 𝑅 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)))
3		dmqmap 38733	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)
4	3	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∈ dom QMap 𝑅 ↔ 𝐴 ∈ dom 𝑅))
5	3	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐵 ∈ dom QMap 𝑅 ↔ 𝐵 ∈ dom 𝑅))
6	4, 5	anbi12d 633	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
7	6	pm5.32i 574	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅)) ↔ (𝑅 ∈ 𝑉 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
8	7	imbi1i 349	. . . . . . 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 38729	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
10		ecqmap 38729	. . . . . . . . . . 11 ⊢ (𝐵 ∈ dom 𝑅 → [𝐵] QMap 𝑅 = {[𝐵]𝑅})
11	9, 10	eqeqan12d 2751	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 ↔ {[𝐴]𝑅} = {[𝐵]𝑅}))
12	11	imbi1d 341	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵) ↔ ({[𝐴]𝑅} = {[𝐵]𝑅} → 𝐴 = 𝐵)))
13		ecexg 8651	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → [𝐴]𝑅 ∈ V)
14		sneqbg 4801	. . . . . . . . . . 11 ⊢ ([𝐴]𝑅 ∈ V → ({[𝐴]𝑅} = {[𝐵]𝑅} ↔ [𝐴]𝑅 = [𝐵]𝑅))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ({[𝐴]𝑅} = {[𝐵]𝑅} ↔ [𝐴]𝑅 = [𝐵]𝑅))
16	15	imbi1d 341	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (({[𝐴]𝑅} = {[𝐵]𝑅} → 𝐴 = 𝐵) ↔ ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
17	12, 16	sylan9bbr 510	. . . . . . . 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 271	. . . . . . 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 275	. . . . . 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 450	. . . . . 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 450	. . . . . 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 301	. . . . 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 272	. . . 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 244	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
25	1, 24	sylbid 240	. 2 ⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
26	25	3imp 1111	1 ⊢ ((𝑅 ∈ 𝑉 ∧ QMap 𝑅 ∈ Disjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  dom cdm 5634  [cec 8645   QMap cqmap 38455   Disjs cdisjs 38498   Disj wdisjALTV 38499

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ec 8649  df-rels 38720  df-qmap 38726  df-coss 38781  df-ssr 38858  df-cnvrefs 38885  df-cnvrefrels 38886  df-cnvrefrel 38887  df-disjss 39068  df-disjs 39069  df-disjALTV 39070

This theorem is referenced by:  qmapeldisjsbi  39141  eldisjs6  39220