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Theorem strfv2d 15682
Description: Deduction version of strfv 15684. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
strfv2d.e 𝐸 = Slot (𝐸‘ndx)
strfv2d.s (𝜑𝑆𝑉)
strfv2d.f (𝜑 → Fun 𝑆)
strfv2d.n (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
strfv2d.c (𝜑𝐶𝑊)
Assertion
Ref Expression
strfv2d (𝜑𝐶 = (𝐸𝑆))

Proof of Theorem strfv2d
StepHypRef Expression
1 strfv2d.e . . 3 𝐸 = Slot (𝐸‘ndx)
2 strfv2d.s . . 3 (𝜑𝑆𝑉)
31, 2strfvnd 15659 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 cnvcnv2 5492 . . . . 5 𝑆 = (𝑆 ↾ V)
54fveq1i 6089 . . . 4 (𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx))
6 fvex 6098 . . . . 5 (𝐸‘ndx) ∈ V
7 fvres 6102 . . . . 5 ((𝐸‘ndx) ∈ V → ((𝑆 ↾ V)‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx)))
86, 7ax-mp 5 . . . 4 ((𝑆 ↾ V)‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx))
95, 8eqtri 2631 . . 3 (𝑆‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx))
10 strfv2d.f . . . 4 (𝜑 → Fun 𝑆)
11 strfv2d.n . . . . . 6 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
12 strfv2d.c . . . . . . . 8 (𝜑𝐶𝑊)
13 elex 3184 . . . . . . . 8 (𝐶𝑊𝐶 ∈ V)
1412, 13syl 17 . . . . . . 7 (𝜑𝐶 ∈ V)
15 opelxpi 5062 . . . . . . 7 (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
166, 14, 15sylancr 693 . . . . . 6 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
1711, 16elind 3759 . . . . 5 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (𝑆 ∩ (V × V)))
18 cnvcnv 5491 . . . . 5 𝑆 = (𝑆 ∩ (V × V))
1917, 18syl6eleqr 2698 . . . 4 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
20 funopfv 6130 . . . 4 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
2110, 19, 20sylc 62 . . 3 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
229, 21syl5eqr 2657 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
233, 22eqtr2d 2644 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cin 3538  cop 4130   × cxp 5026  ccnv 5027  cres 5030  Fun wfun 5784  cfv 5790  ndxcnx 15641  Slot cslot 15643
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-res 5040  df-iota 5754  df-fun 5792  df-fv 5798  df-slot 15648
This theorem is referenced by:  strfv2  15683  eengbas  25607  ebtwntg  25608  ecgrtg  25609  elntg  25610
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