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Theorem strfvd 16111
 Description: Deduction version of strfv 16114. (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 16083 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 strfvd.f . . 3 (𝜑 → Fun 𝑆)
5 strfvd.n . . 3 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
6 funopfv 6376 . . 3 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
74, 5, 6sylc 65 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
83, 7eqtr2d 2806 1 (𝜑𝐶 = (𝐸𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ⟨cop 4322  Fun wfun 6025  ‘cfv 6031  ndxcnx 16061  Slot cslot 16063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-slot 16068 This theorem is referenced by:  strssd  16116
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