Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem11 Structured version   Visualization version   GIF version

Theorem prtlem11 38370
Description: Lemma for prter2 38385. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
prtlem11 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))

Proof of Theorem prtlem11
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eceq1 8769 . . . 4 (𝑥 = 𝐶 → [𝑥] = [𝐶] )
21rspceeqv 3633 . . 3 ((𝐶𝐴𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
3 elqsg 8793 . . 3 (𝐵𝐷 → (𝐵 ∈ (𝐴 / ) ↔ ∃𝑥𝐴 𝐵 = [𝑥] ))
42, 3imbitrrid 245 . 2 (𝐵𝐷 → ((𝐶𝐴𝐵 = [𝐶] ) → 𝐵 ∈ (𝐴 / )))
54expd 414 1 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wrex 3067  [cec 8729   / cqs 8730
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-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737
This theorem is referenced by:  prter2  38385
  Copyright terms: Public domain W3C validator