Theorem disjimeceqim2 39929

Description: Disj implies injectivity (pairwise form). The same content as disjimeceqim 39928 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 783	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → 𝐴 ∈ dom 𝑅)
2		simprr 785	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → 𝐵 ∈ dom 𝑅)
3		eleq1 2852	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∈ dom 𝑅 ↔ 𝐴 ∈ dom 𝑅))
4		eleq1 2852	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅))
5	3, 4	bi2anan9 650	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
6		eceq1 8785	. . . . . . 7 ⊢ (𝑢 = 𝐴 → [𝑢]𝑅 = [𝐴]𝑅)
7		eceq1 8785	. . . . . . 7 ⊢ (𝑣 = 𝐵 → [𝑣]𝑅 = [𝐵]𝑅)
8	6, 7	eqeqan12d 2777	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ([𝑢]𝑅 = [𝑣]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅))
9		eqeq12 2780	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑢 = 𝑣 ↔ 𝐴 = 𝐵))
10	8, 9	imbi12d 347	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) ↔ ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
11	5, 10	imbi12d 347	. . . 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 39928	. . . . . 6 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
13		rsp2 3283	. . . . . 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 18	. . . . 5 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣)))
15	14	adantr 486	. . . 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 3527	. . 3 ⊢ (( Disj 𝑅 ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
17	16	ex 418	. 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 54	1 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  dom cdm 5655  [cec 8743   Disj wdisjALTV 39343

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8747  df-coss 39625  df-cnvrefrel 39731  df-disjALTV 39914

This theorem is used by:  disjimeceqbi2  39931  qmapeldisjsim  39984