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Theorem strfvd 15898
Description: Deduction version of strfv 15901. (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 15870 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 strfvd.f . . 3 (𝜑 → Fun 𝑆)
5 strfvd.n . . 3 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
6 funopfv 6233 . . 3 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
74, 5, 6sylc 65 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
83, 7eqtr2d 2656 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  cop 4181  Fun wfun 5880  cfv 5886  ndxcnx 15848  Slot cslot 15850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-slot 15855
This theorem is referenced by:  strssd  15903
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