Theorem qmapeldisjsim 39181

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 39146	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))
2		disjimeceqim2 39126	. . . 4 ⊢ ( Disj QMap 𝑅 → ((𝐴 ∈ dom QMap 𝑅 ∧ 𝐵 ∈ dom QMap 𝑅) → ([𝐴] QMap 𝑅 = [𝐵] QMap 𝑅 → 𝐴 = 𝐵)))
3		dmqmap 38774	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)
4	3	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∈ dom QMap 𝑅 ↔ 𝐴 ∈ dom 𝑅))
5	3	eleq2d 2822	. . . . . . . . . 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 38770	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})
10		ecqmap 38770	. . . . . . . . . . 11 ⊢ (𝐵 ∈ dom 𝑅 → [𝐵] QMap 𝑅 = {[𝐵]𝑅})
11	9, 10	eqeqan12d 2750	. . . . . . . . . 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 8647	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → [𝐴]𝑅 ∈ V)
14		sneqbg 4786	. . . . . . . . . . 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 3429  {csn 4567  dom cdm 5631  [cec 8641   QMap cqmap 38496   Disjs cdisjs 38539   Disj wdisjALTV 38540

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ec 8645  df-rels 38761  df-qmap 38767  df-coss 38822  df-ssr 38899  df-cnvrefs 38926  df-cnvrefrels 38927  df-cnvrefrel 38928  df-disjss 39109  df-disjs 39110  df-disjALTV 39111

This theorem is referenced by:  qmapeldisjsbi  39182  eldisjs6  39261