Theorem disjimeceqim2 39085

Description: Disj implies injectivity (pairwise form). The same content as disjimeceqim 39084 but packaged for direct use with explicit hypotheses (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅). (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
disjimeceqim2	⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))

Proof of Theorem disjimeceqim2

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 771	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → 𝐴 ∈ dom 𝑅)
2		simprr 773	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → 𝐵 ∈ dom 𝑅)
3		eleq1 2825	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
4		eleq1 2825	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅))
5	3, 4	bi2anan9 639	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
6		eceq1 8687	. . . . . . 7 ⊢ (𝑢 = 𝐴 → [𝑢]𝑅 = [𝐴]𝑅)
7		eceq1 8687	. . . . . . 7 ⊢ (𝑣 = 𝐵 → [𝑣]𝑅 = [𝐵]𝑅)
8	6, 7	eqeqan12d 2751	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ([𝑢]𝑅 = [𝑣]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅))
9		eqeq12 2754	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑢 = 𝑣 ↔ 𝐴 = 𝐵))
10	8, 9	imbi12d 344	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) ↔ ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
11	5, 10	imbi12d 344	. . . 4 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣)) ↔ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
12		disjimeceqim 39084	. . . . . 6 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
13		rsp2 3255	. . . . . 6 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣)))
14	12, 13	syl 17	. . . . 5 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣)))
15	14	adantr 480	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣)))
16	1, 2, 11, 15	vtocl2d 3521	. . 3 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
17	16	ex 412	. 2 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))))
18	17	pm2.43d 53	1 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  dom cdm 5634  [cec 8645   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-sep 5245  ax-pr 5381

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-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  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-ec 8649  df-coss 38781  df-cnvrefrel 38887  df-disjALTV 39070

This theorem is referenced by:  disjimeceqbi2  39087  qmapeldisjsim  39140