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Theorem ov2ssiunov2 43713
Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15096 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 1139 . . . 4 ((𝑅𝑈𝑁𝑉𝑀𝑁) → 𝑀𝑁)
2 simpr 484 . . . . . 6 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
32oveq2d 7447 . . . . 5 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑅 𝑛) = (𝑅 𝑀))
43eleq2d 2827 . . . 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 43692 . . . . 5 ((𝑅𝑈𝑁𝑉) → (𝑥 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
87biimprd 248 . . . 4 ((𝑅𝑈𝑁𝑉) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
983adant3 1133 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
105, 9syld 47 . 2 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → 𝑥 ∈ (𝐶𝑅)))
1110ssrdv 3989 1 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wss 3951   ciun 4991  cmpt 5225  cfv 6561  (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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434
This theorem is referenced by:  dftrcl3  43733  dfrtrcl3  43746
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