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Theorem ov2ssiunov2 43690
Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15093 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
Hypothesis
Ref Expression
ov2ssiunov2.def 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
Assertion
Ref Expression
ov2ssiunov2 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
Distinct variable groups:   𝑛,𝑟,𝐶,𝑁,   𝑛,𝑀   𝑅,𝑟,𝑛   𝑈,𝑛   𝑛,𝑉
Allowed substitution hints:   𝑈(𝑟)   𝑀(𝑟)   𝑉(𝑟)

Proof of Theorem ov2ssiunov2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . . 4 ((𝑅𝑈𝑁𝑉𝑀𝑁) → 𝑀𝑁)
2 simpr 484 . . . . . 6 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
32oveq2d 7447 . . . . 5 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑅 𝑛) = (𝑅 𝑀))
43eleq2d 2825 . . . 4 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 𝑛) ↔ 𝑥 ∈ (𝑅 𝑀)))
51, 4rspcedv 3615 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
6 ov2ssiunov2.def . . . . . 6 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
76eliunov2 43669 . . . . 5 ((𝑅𝑈𝑁𝑉) → (𝑥 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
87biimprd 248 . . . 4 ((𝑅𝑈𝑁𝑉) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
983adant3 1131 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
105, 9syld 47 . 2 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → 𝑥 ∈ (𝐶𝑅)))
1110ssrdv 4001 1 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963   ciun 4996  cmpt 5231  cfv 6563  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434
This theorem is referenced by:  dftrcl3  43710  dfrtrcl3  43723
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