Theorem eldisjdmqsim 39750

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 4093	. . . 4 ⊢ (𝑥 ∈ ([𝑢]𝑅 ∩ [𝑣]𝑅) ↔ (𝑥 ∈ [𝑢]𝑅 ∧ 𝑥 ∈ [𝑣]𝑅))
2		elecALTV 39204	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥))
3	2	el2v 3457	. . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)
4		elecALTV 39204	. . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑣]𝑅 ↔ 𝑣𝑅𝑥))
5	4	el2v 3457	. . . . 5 ⊢ (𝑥 ∈ [𝑣]𝑅 ↔ 𝑣𝑅𝑥)
6	3, 5	anbi12i 640	. . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑥 ∈ [𝑣]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))
7	1, 6	bitr2i 279	. . 3 ⊢ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) ↔ 𝑥 ∈ ([𝑢]𝑅 ∩ [𝑣]𝑅))
8		ne0i 4286	. . 3 ⊢ (𝑥 ∈ ([𝑢]𝑅 ∩ [𝑣]𝑅) → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
9	7, 8	sylbi 220	. 2 ⊢ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
10		19.8a 2217	. . . . . . 7 ⊢ (𝑢𝑅𝑥 → ∃𝑥 𝑢𝑅𝑥)
11		eldmg 5879	. . . . . . . 8 ⊢ (𝑢 ∈ V → (𝑢 ∈ dom 𝑅 ↔ ∃𝑥 𝑢𝑅𝑥))
12	11	elv 3455	. . . . . . 7 ⊢ (𝑢 ∈ dom 𝑅 ↔ ∃𝑥 𝑢𝑅𝑥)
13	10, 12	sylibr 237	. . . . . 6 ⊢ (𝑢𝑅𝑥 → 𝑢 ∈ dom 𝑅)
14		19.8a 2217	. . . . . . 7 ⊢ (𝑣𝑅𝑥 → ∃𝑥 𝑣𝑅𝑥)
15		eldmg 5879	. . . . . . . 8 ⊢ (𝑣 ∈ V → (𝑣 ∈ dom 𝑅 ↔ ∃𝑥 𝑣𝑅𝑥))
16	15	elv 3455	. . . . . . 7 ⊢ (𝑣 ∈ dom 𝑅 ↔ ∃𝑥 𝑣𝑅𝑥)
17	14, 16	sylibr 237	. . . . . 6 ⊢ (𝑣𝑅𝑥 → 𝑣 ∈ dom 𝑅)
18	13, 17	anim12i 625	. . . . 5 ⊢ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅))
19		eceldmqs 8808	. . . . . 6 ⊢ (𝑅 ∈ Rels → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑢 ∈ dom 𝑅))
20		eceldmqs 8808	. . . . . 6 ⊢ (𝑅 ∈ Rels → ([𝑣]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑣 ∈ dom 𝑅))
21	19, 20	anbi12d 644	. . . . 5 ⊢ (𝑅 ∈ Rels → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)))
22	18, 21	imbitrrid 249	. . . 4 ⊢ (𝑅 ∈ Rels → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅))))
23	22	adantl 487	. . 3 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅))))
24		eldisjim3 39748	. . . 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 486	. . 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 48	. 2 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
27	9, 26	mpdi 46	1 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → [𝑢]𝑅 = [𝑣]𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ∩ cin 4089  ∅c0 4278   class class class wbr 5102  dom cdm 5650  [cec 8715   / cqs 8716   Rels crels 39118   ElDisj weldisj 39154

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8719  df-qs 8723  df-coss 39434  df-cnvrefrel 39540  df-disjALTV 39723  df-eldisj 39725

This theorem is used by: (None)