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Theorem disjdmqscossss 37542
Description: Lemma for disjdmqseq 37544 via disjdmqs 37543. (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.
StepHypRef Expression
1 disjrel 37469 . . . . . . . 8 ( Disj 𝑅 → Rel 𝑅)
2 releldmqscoss 37399 . . . . . . . . 9 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
32elv 3480 . . . . . . . 8 (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
41, 3syl 17 . . . . . . 7 ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
5 disjlem19 37540 . . . . . . . . . 10 (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
65elv 3480 . . . . . . . . 9 ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
76ralrimivv 3198 . . . . . . . 8 ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8 2r19.29 3139 . . . . . . . . 9 ((∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅))
98ex 413 . . . . . . . 8 (∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅 → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅)))
107, 9syl 17 . . . . . . 7 ( Disj 𝑅 → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅 → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅)))
114, 10sylbid 239 . . . . . 6 ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅)))
12 eqtr3 2758 . . . . . . . 8 (([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅) → [𝑢]𝑅 = 𝑣)
1312reximi 3084 . . . . . . 7 (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
1413reximi 3084 . . . . . 6 (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑥] ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
1511, 14syl6 35 . . . . 5 ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣))
16 df-rex 3071 . . . . . . . 8 (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
17 19.41v 1953 . . . . . . . 8 (∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
1816, 17bitri 274 . . . . . . 7 (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
1918simprbi 497 . . . . . 6 (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → [𝑢]𝑅 = 𝑣)
2019reximi 3084 . . . . 5 (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣)
2115, 20syl6 35 . . . 4 ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣))
22 eqcom 2739 . . . . 5 ([𝑢]𝑅 = 𝑣𝑣 = [𝑢]𝑅)
2322rexbii 3094 . . . 4 (∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅)
2421, 23syl6ib 250 . . 3 ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) → ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅))
2524ss2abdv 4057 . 2 ( Disj 𝑅 → {𝑣𝑣 ∈ (dom ≀ 𝑅 /𝑅)} ⊆ {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅})
26 abid1 2870 . 2 (dom ≀ 𝑅 /𝑅) = {𝑣𝑣 ∈ (dom ≀ 𝑅 /𝑅)}
27 df-qs 8694 . 2 (dom 𝑅 / 𝑅) = {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅}
2825, 26, 273sstr4g 4024 1 ( Disj 𝑅 → (dom ≀ 𝑅 /𝑅) ⊆ (dom 𝑅 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wral 3061  wrex 3070  Vcvv 3474  wss 3945  dom cdm 5670  Rel wrel 5675  [cec 8686   / cqs 8687  ccoss 36912   Disj wdisjALTV 36946
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ec 8690  df-qs 8694  df-coss 37150  df-cnvrefrel 37266  df-disjALTV 37444
This theorem is referenced by:  disjdmqs  37543
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