MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabss3d Structured version   Visualization version   GIF version

Theorem rabss3d 4077
Description: Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
Hypothesis
Ref Expression
rabss3d.1 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
Assertion
Ref Expression
rabss3d (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabss3d
StepHypRef Expression
1 nfv 1909 . 2 𝑥𝜑
2 nfrab1 3438 . 2 𝑥{𝑥𝐴𝜓}
3 nfrab1 3438 . 2 𝑥{𝑥𝐵𝜓}
4 rabss3d.1 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
5 simprr 771 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜓)
64, 5jca 510 . . . 4 ((𝜑 ∧ (𝑥𝐴𝜓)) → (𝑥𝐵𝜓))
76ex 411 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜓)))
8 rabid 3439 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
9 rabid 3439 . . 3 (𝑥 ∈ {𝑥𝐵𝜓} ↔ (𝑥𝐵𝜓))
107, 8, 93imtr4g 295 . 2 (𝜑 → (𝑥 ∈ {𝑥𝐴𝜓} → 𝑥 ∈ {𝑥𝐵𝜓}))
111, 2, 3, 10ssrd 3983 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  {crab 3418  wss 3946
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 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-rab 3419  df-ss 3963
This theorem is referenced by:  xpinpreima2  33690  reprss  34431  reprinfz1  34436  upwrdfi  46455
  Copyright terms: Public domain W3C validator