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

Theorem strfvd 16311
Description: Deduction version of strfv 16314. (Contributed by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
strfvd.e 𝐸 = Slot (𝐸‘ndx)
strfvd.s (𝜑𝑆𝑉)
strfvd.f (𝜑 → Fun 𝑆)
strfvd.n (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
Assertion
Ref Expression
strfvd (𝜑𝐶 = (𝐸𝑆))

Proof of Theorem strfvd
StepHypRef Expression
1 strfvd.e . . 3 𝐸 = Slot (𝐸‘ndx)
2 strfvd.s . . 3 (𝜑𝑆𝑉)
31, 2strfvnd 16285 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 strfvd.f . . 3 (𝜑 → Fun 𝑆)
5 strfvd.n . . 3 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
6 funopfv 6496 . . 3 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
74, 5, 6sylc 65 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
83, 7eqtr2d 2815 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cop 4404  Fun wfun 6131  cfv 6137  ndxcnx 16263  Slot cslot 16265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-slot 16270
This theorem is referenced by:  strssd  16316
  Copyright terms: Public domain W3C validator