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Theorem ov2ssiunov2 37011
 Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 13646 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 1056 . . . 4 ((𝑅𝑈𝑁𝑉𝑀𝑁) → 𝑀𝑁)
2 simpr 476 . . . . . 6 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
32oveq2d 6565 . . . . 5 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑅 𝑛) = (𝑅 𝑀))
43eleq2d 2673 . . . 4 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 𝑛) ↔ 𝑥 ∈ (𝑅 𝑀)))
51, 4rspcedv 3286 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
6 ov2ssiunov2.def . . . . . 6 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
76eliunov2 36990 . . . . 5 ((𝑅𝑈𝑁𝑉) → (𝑥 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
87biimprd 237 . . . 4 ((𝑅𝑈𝑁𝑉) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
983adant3 1074 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
105, 9syld 46 . 2 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → 𝑥 ∈ (𝐶𝑅)))
1110ssrdv 3574 1 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ ciun 4455   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552 This theorem is referenced by:  dftrcl3  37031  dfrtrcl3  37044
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