Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ov2ssiunov2 Structured version   Visualization version   GIF version

Theorem ov2ssiunov2 43032
Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15010 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 1135 . . . 4 ((𝑅𝑈𝑁𝑉𝑀𝑁) → 𝑀𝑁)
2 simpr 484 . . . . . 6 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
32oveq2d 7421 . . . . 5 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑅 𝑛) = (𝑅 𝑀))
43eleq2d 2813 . . . 4 (((𝑅𝑈𝑁𝑉𝑀𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 𝑛) ↔ 𝑥 ∈ (𝑅 𝑀)))
51, 4rspcedv 3599 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
6 ov2ssiunov2.def . . . . . 6 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
76eliunov2 43011 . . . . 5 ((𝑅𝑈𝑁𝑉) → (𝑥 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛)))
87biimprd 247 . . . 4 ((𝑅𝑈𝑁𝑉) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
983adant3 1129 . . 3 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (∃𝑛𝑁 𝑥 ∈ (𝑅 𝑛) → 𝑥 ∈ (𝐶𝑅)))
105, 9syld 47 . 2 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑥 ∈ (𝑅 𝑀) → 𝑥 ∈ (𝐶𝑅)))
1110ssrdv 3983 1 ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  wss 3943   ciun 4990  cmpt 5224  cfv 6537  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408
This theorem is referenced by:  dftrcl3  43052  dfrtrcl3  43065
  Copyright terms: Public domain W3C validator