Theorem disjdmqscossss 39575

Description: Lemma for disjdmqseq 39577 via disjdmqs 39576. (Contributed by Peter Mazsa, 16-Sep-2021.)

Assertion

Ref	Expression
disjdmqscossss	⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))

Proof of Theorem disjdmqscossss

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

Step	Hyp	Ref	Expression
1		disjrel 39499	. . . . . . . 8 ⊢ ( Disj 𝑅 → Rel 𝑅)
2		releldmqscoss 39414	. . . . . . . . 9 ⊢ (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
3	2	elv 3460	. . . . . . . 8 ⊢ (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
4	1, 3	syl 18	. . . . . . 7 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
5		disjlem19 39573	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
6	5	elv 3460	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
7	6	ralrimivv 3206	. . . . . . . 8 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8		2r19.29 3151	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅))
9	8	ex 417	. . . . . . . 8 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅 → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
10	7, 9	syl 18	. . . . . . 7 ⊢ ( Disj 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅 → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
11	4, 10	sylbid 243	. . . . . 6 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
12		eqtr3 2785	. . . . . . . 8 ⊢ (([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → [𝑢]𝑅 = 𝑣)
13	12	reximi 3103	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
14	13	reximi 3103	. . . . . 6 ⊢ (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
15	11, 14	syl6 36	. . . . 5 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣))
16		df-rex 3090	. . . . . . . 8 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
17		19.41v 1979	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
18	16, 17	bitri 278	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
19	18	simprbi 502	. . . . . 6 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → [𝑢]𝑅 = 𝑣)
20	19	reximi 3103	. . . . 5 ⊢ (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣)
21	15, 20	syl6 36	. . . 4 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣))
22		eqcom 2770	. . . . 5 ⊢ ([𝑢]𝑅 = 𝑣 ↔ 𝑣 = [𝑢]𝑅)
23	22	rexbii 3112	. . . 4 ⊢ (∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅)
24	21, 23	imbitrdi 254	. . 3 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅))
25	24	ss2abdv 4019	. 2 ⊢ ( Disj 𝑅 → {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)} ⊆ {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅})
26		abid1 2899	. 2 ⊢ (dom ≀ 𝑅 / ≀ 𝑅) = {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)}
27		df-qs 8696	. 2 ⊢ (dom 𝑅 / 𝑅) = {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅}
28	25, 26, 27	3sstr4g 3990	1 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  dom cdm 5661  Rel wrel 5666  [cec 8688   / cqs 8689   ≀ ccoss 38852   Disj wdisjALTV 38888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-coss 39170  df-cnvrefrel 39276  df-disjALTV 39459

This theorem is referenced by:  disjdmqs  39576