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Theorem cllem0 38373
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 Richard Penner, 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 3353 . . . . . 6 𝑅 ∈ V
3 cllem0.r . . . . . 6 (𝑧 = 𝑅 → (𝜑𝜓))
4 cllem0.v . . . . . 6 𝑉 = {𝑧𝜑}
52, 3, 4elab2 3494 . . . . 5 (𝑅𝑉𝜓)
65ralbii 3118 . . . 4 (∀𝑦𝑉 𝑅𝑉 ↔ ∀𝑦𝑉 𝜓)
76ralbii 3118 . . 3 (∀𝑥𝑉𝑦𝑉 𝑅𝑉 ↔ ∀𝑥𝑉𝑦𝑉 𝜓)
8 df-ral 3055 . . . 4 (∀𝑦𝑉 𝜓 ↔ ∀𝑦(𝑦𝑉𝜓))
98ralbii 3118 . . 3 (∀𝑥𝑉𝑦𝑉 𝜓 ↔ ∀𝑥𝑉𝑦(𝑦𝑉𝜓))
10 df-ral 3055 . . 3 (∀𝑥𝑉𝑦(𝑦𝑉𝜓) ↔ ∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑉𝜓)))
117, 9, 103bitri 286 . 2 (∀𝑥𝑉𝑦𝑉 𝑅𝑉 ↔ ∀𝑥(𝑥𝑉 → ∀𝑦(𝑦𝑉𝜓)))
12 vex 3343 . . . . . 6 𝑥 ∈ V
13 cllem0.x . . . . . 6 (𝑧 = 𝑥 → (𝜑𝜒))
1412, 13, 4elab2 3494 . . . . 5 (𝑥𝑉𝜒)
15 vex 3343 . . . . . 6 𝑦 ∈ V
16 cllem0.y . . . . . 6 (𝑧 = 𝑦 → (𝜑𝜃))
1715, 16, 4elab2 3494 . . . . 5 (𝑦𝑉𝜃)
18 cllem0.closed . . . . 5 ((𝜒𝜃) → 𝜓)
1914, 17, 18syl2anb 497 . . . 4 ((𝑥𝑉𝑦𝑉) → 𝜓)
2019ex 449 . . 3 (𝑥𝑉 → (𝑦𝑉𝜓))
2120alrimiv 2004 . 2 (𝑥𝑉 → ∀𝑦(𝑦𝑉𝜓))
2211, 21mpgbir 1875 1 𝑥𝑉𝑦𝑉 𝑅𝑉
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342
This theorem is referenced by:  superficl  38374  superuncl  38375  ssficl  38376  ssuncl  38377  ssdifcl  38378  sssymdifcl  38379  trficl  38463
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