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Theorem ov2ssiunov2 39405
Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14278 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 1118 . . . 4 ((𝑅𝑈𝑁𝑉𝑀𝑁) → 𝑀𝑁)
2 simpr 477 . . . . . 6 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
32oveq2d 6992 . . . . 5 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑅 𝑛) = (𝑅 𝑀))
43eleq2d 2852 . . . 4 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 𝑛) ↔ 𝑥 ∈ (𝑅 𝑀)))
51, 4rspcedv 3540 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
6 ov2ssiunov2.def . . . . . 6 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
76eliunov2 39384 . . . . 5 ((𝑅𝑈𝑁𝑉) → (𝑥 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
87biimprd 240 . . . 4 ((𝑅𝑈𝑁𝑉) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
983adant3 1112 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
105, 9syld 47 . 2 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → 𝑥 ∈ (𝐶𝑅)))
1110ssrdv 3865 1 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3090  Vcvv 3416  wss 3830   ciun 4792  cmpt 5008  cfv 6188  (class class class)co 6976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979
This theorem is referenced by:  dftrcl3  39425  dfrtrcl3  39438
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