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Theorem strfv2d 15952
Description: Deduction version of strfv 15954. (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 15923 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 cnvcnv2 5623 . . . . 5 𝑆 = (𝑆 ↾ V)
54fveq1i 6230 . . . 4 (𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx))
6 fvex 6239 . . . . 5 (𝐸‘ndx) ∈ V
7 fvres 6245 . . . . 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 2673 . . 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 3243 . . . . . . . 8 (𝐶𝑊𝐶 ∈ V)
1412, 13syl 17 . . . . . . 7 (𝜑𝐶 ∈ V)
15 opelxpi 5182 . . . . . . 7 (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
166, 14, 15sylancr 696 . . . . . 6 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
1711, 16elind 3831 . . . . 5 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (𝑆 ∩ (V × V)))
18 cnvcnv 5621 . . . . 5 𝑆 = (𝑆 ∩ (V × V))
1917, 18syl6eleqr 2741 . . . 4 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
20 funopfv 6273 . . . 4 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
2110, 19, 20sylc 65 . . 3 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
229, 21syl5eqr 2699 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
233, 22eqtr2d 2686 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  cop 4216   × cxp 5141  ccnv 5142  cres 5145  Fun wfun 5920  cfv 5926  ndxcnx 15901  Slot cslot 15903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-slot 15908
This theorem is referenced by:  strfv2  15953  opelstrbas  16025  eengbas  25906  ebtwntg  25907  ecgrtg  25908  elntg  25909  edgfiedgval  25947
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