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Theorem strfv2d 16529
Description: Deduction version of strfv2 16530. (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 16502 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 cnvcnv2 6037 . . . . 5 𝑆 = (𝑆 ↾ V)
54fveq1i 6662 . . . 4 (𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx))
6 fvex 6674 . . . . 5 (𝐸‘ndx) ∈ V
7 fvres 6680 . . . . 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 2847 . . 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)
1312elexd 3500 . . . . . . 7 (𝜑𝐶 ∈ V)
14 opelxpi 5579 . . . . . . 7 (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
156, 13, 14sylancr 590 . . . . . 6 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
1611, 15elind 4156 . . . . 5 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (𝑆 ∩ (V × V)))
17 cnvcnv 6036 . . . . 5 𝑆 = (𝑆 ∩ (V × V))
1816, 17eleqtrrdi 2927 . . . 4 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
19 funopfv 6708 . . . 4 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
2010, 18, 19sylc 65 . . 3 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
219, 20syl5eqr 2873 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
223, 21eqtr2d 2860 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918  cop 4556   × cxp 5540  ccnv 5541  cres 5544  Fun wfun 6337  cfv 6343  ndxcnx 16480  Slot cslot 16482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-slot 16487
This theorem is referenced by:  strfv2  16530  opelstrbas  16597  ebtwntg  26779  ecgrtg  26780  elntg  26781  edgfiedgval  26813
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