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Theorem strfvd 15678
Description: Deduction version of strfv 15681. (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 15656 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 strfvd.f . . 3 (𝜑 → Fun 𝑆)
5 strfvd.n . . 3 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
6 funopfv 6130 . . 3 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
74, 5, 6sylc 62 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
83, 7eqtr2d 2644 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cop 4130  Fun wfun 5784  cfv 5790  ndxcnx 15638  Slot cslot 15640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-slot 15645
This theorem is referenced by:  strssd  15683
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