Theorem eldisjdmqsim 39097

Description: Shared output implies equal cosets (under ElDisj of quotient): if 𝑢 and 𝑣 both relate to the same 𝑥, then their cosets intersect, hence must coincide under quotient ElDisj. (Contributed by Peter Mazsa, 10-Feb-2026.)

Assertion

Ref	Expression
eldisjdmqsim	⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → [𝑢]𝑅 = [𝑣]𝑅))

Distinct variable groups:   𝑥,𝑅   𝑥,𝑢   𝑥,𝑣

Allowed substitution hints:   𝑅(𝑣,𝑢)

Proof of Theorem eldisjdmqsim

Step	Hyp	Ref	Expression
1		elin 3919	. . . 4 ⊢ (𝑥 ∈ ([𝑢]𝑅 ∩ [𝑣]𝑅) ↔ (𝑥 ∈ [𝑢]𝑅 ∧ 𝑥 ∈ [𝑣]𝑅))
2		elecALTV 38551	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥))
3	2	el2v 3449	. . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)
4		elecALTV 38551	. . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑣]𝑅 ↔ 𝑣𝑅𝑥))
5	4	el2v 3449	. . . . 5 ⊢ (𝑥 ∈ [𝑣]𝑅 ↔ 𝑣𝑅𝑥)
6	3, 5	anbi12i 629	. . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑥 ∈ [𝑣]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))
7	1, 6	bitr2i 276	. . 3 ⊢ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) ↔ 𝑥 ∈ ([𝑢]𝑅 ∩ [𝑣]𝑅))
8		ne0i 4295	. . 3 ⊢ (𝑥 ∈ ([𝑢]𝑅 ∩ [𝑣]𝑅) → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
9	7, 8	sylbi 217	. 2 ⊢ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
10		19.8a 2189	. . . . . . 7 ⊢ (𝑢𝑅𝑥 → ∃𝑥 𝑢𝑅𝑥)
11		eldmg 5857	. . . . . . . 8 ⊢ (𝑢 ∈ V → (𝑢 ∈ dom 𝑅 ↔ ∃𝑥 𝑢𝑅𝑥))
12	11	elv 3447	. . . . . . 7 ⊢ (𝑢 ∈ dom 𝑅 ↔ ∃𝑥 𝑢𝑅𝑥)
13	10, 12	sylibr 234	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑢 ∈ dom 𝑅)
14		19.8a 2189	. . . . . . 7 ⊢ (𝑣𝑅𝑥 → ∃𝑥 𝑣𝑅𝑥)
15		eldmg 5857	. . . . . . . 8 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom 𝑅 ↔ ∃𝑥 𝑣𝑅𝑥))
16	15	elv 3447	. . . . . . 7 ⊢ (𝑣 ∈ dom 𝑅 ↔ ∃𝑥 𝑣𝑅𝑥)
17	14, 16	sylibr 234	. . . . . 6 ⊢ (𝑣𝑅𝑥 → 𝑣 ∈ dom 𝑅)
18	13, 17	anim12i 614	. . . . 5 ⊢ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅))
19		eceldmqs 8738	. . . . . 6 ⊢ (𝑅 ∈ Rels → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑢 ∈ dom 𝑅))
20		eceldmqs 8738	. . . . . 6 ⊢ (𝑅 ∈ Rels → ([𝑣]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑣 ∈ dom 𝑅))
21	19, 20	anbi12d 633	. . . . 5 ⊢ (𝑅 ∈ Rels → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)))
22	18, 21	imbitrrid 246	. . . 4 ⊢ (𝑅 ∈ Rels → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅))))
23	22	adantl 481	. . 3 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅))))
24		eldisjim3 39095	. . . 4 ⊢ ( ElDisj (dom 𝑅 / 𝑅) → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
25	24	adantr 480	. . 3 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
26	23, 25	syld 47	. 2 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
27	9, 26	mpdi 45	1 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → [𝑢]𝑅 = [𝑣]𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∩ cin 3902  ∅c0 4287   class class class wbr 5100  dom cdm 5634  [cec 8645   / cqs 8646   Rels crels 38465   ElDisj weldisj 38501

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  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-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-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-eprel 5534  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-qs 8653  df-coss 38781  df-cnvrefrel 38887  df-disjALTV 39070  df-eldisj 39072

This theorem is referenced by: (None)