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Theorem cllem0 40638
 Description: The class of all sets with property 𝜑(𝑧) is closed under the binary operation on sets defined in 𝑅(𝑥, 𝑦). (Contributed by RP, 3-Jan-2020.)
Hypotheses
Ref Expression
cllem0.v 𝑉 = {𝑧𝜑}
cllem0.rex 𝑅𝑈
cllem0.r (𝑧 = 𝑅 → (𝜑𝜓))
cllem0.x (𝑧 = 𝑥 → (𝜑𝜒))
cllem0.y (𝑧 = 𝑦 → (𝜑𝜃))
cllem0.closed ((𝜒𝜃) → 𝜓)
Assertion
Ref Expression
cllem0 𝑥𝑉𝑦𝑉 𝑅𝑉
Distinct variable groups:   𝜓,𝑧   𝜒,𝑧   𝜃,𝑧   𝑥,𝑦,𝑧   𝑦,𝑉   𝑧,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem cllem0
StepHypRef Expression
1 cllem0.rex . . . . . . 7 𝑅𝑈
21elexi 3429 . . . . . 6 𝑅 ∈ V
3 cllem0.r . . . . . 6 (𝑧 = 𝑅 → (𝜑𝜓))
4 cllem0.v . . . . . 6 𝑉 = {𝑧𝜑}
52, 3, 4elab2 3591 . . . . 5 (𝑅𝑉𝜓)
65ralbii 3097 . . . 4 (∀𝑦𝑉 𝑅𝑉 ↔ ∀𝑦𝑉 𝜓)
76ralbii 3097 . . 3 (∀𝑥𝑉𝑦𝑉 𝑅𝑉 ↔ ∀𝑥𝑉𝑦𝑉 𝜓)
8 df-ral 3075 . . . 4 (∀𝑦𝑉 𝜓 ↔ ∀𝑦(𝑦𝑉𝜓))
98ralbii 3097 . . 3 (∀𝑥𝑉𝑦𝑉 𝜓 ↔ ∀𝑥𝑉𝑦(𝑦𝑉𝜓))
10 df-ral 3075 . . 3 (∀𝑥𝑉𝑦(𝑦𝑉𝜓) ↔ ∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑉𝜓)))
117, 9, 103bitri 300 . 2 (∀𝑥𝑉𝑦𝑉 𝑅𝑉 ↔ ∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑉𝜓)))
12 vex 3413 . . . . . 6 𝑥 ∈ V
13 cllem0.x . . . . . 6 (𝑧 = 𝑥 → (𝜑𝜒))
1412, 13, 4elab2 3591 . . . . 5 (𝑥𝑉𝜒)
15 vex 3413 . . . . . 6 𝑦 ∈ V
16 cllem0.y . . . . . 6 (𝑧 = 𝑦 → (𝜑𝜃))
1715, 16, 4elab2 3591 . . . . 5 (𝑦𝑉𝜃)
18 cllem0.closed . . . . 5 ((𝜒𝜃) → 𝜓)
1914, 17, 18syl2anb 600 . . . 4 ((𝑥𝑉𝑦𝑉) → 𝜓)
2019ex 416 . . 3 (𝑥𝑉 → (𝑦𝑉𝜓))
2120alrimiv 1928 . 2 (𝑥𝑉 → ∀𝑦(𝑦𝑉𝜓))
2211, 21mpgbir 1801 1 𝑥𝑉𝑦𝑉 𝑅𝑉
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411 This theorem is referenced by:  superficl  40639  superuncl  40640  ssficl  40641  ssuncl  40642  ssdifcl  40643  sssymdifcl  40644  trficl  40743
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