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Theorem strfv2d 16111
Description: Deduction version of strfv 16113. (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 16082 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 cnvcnv2 5729 . . . . 5 𝑆 = (𝑆 ↾ V)
54fveq1i 6333 . . . 4 (𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx))
6 fvex 6342 . . . . 5 (𝐸‘ndx) ∈ V
7 fvres 6348 . . . . 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 2792 . . 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 3361 . . . . . . . 8 (𝐶𝑊𝐶 ∈ V)
1412, 13syl 17 . . . . . . 7 (𝜑𝐶 ∈ V)
15 opelxpi 5288 . . . . . . 7 (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
166, 14, 15sylancr 567 . . . . . 6 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
1711, 16elind 3947 . . . . 5 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (𝑆 ∩ (V × V)))
18 cnvcnv 5727 . . . . 5 𝑆 = (𝑆 ∩ (V × V))
1917, 18syl6eleqr 2860 . . . 4 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
20 funopfv 6376 . . . 4 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
2110, 19, 20sylc 65 . . 3 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
229, 21syl5eqr 2818 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
233, 22eqtr2d 2805 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  Vcvv 3349  cin 3720  cop 4320   × cxp 5247  ccnv 5248  cres 5251  Fun wfun 6025  cfv 6031  ndxcnx 16060  Slot cslot 16062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-slot 16067
This theorem is referenced by:  strfv2  16112  opelstrbas  16185  eengbas  26081  ebtwntg  26082  ecgrtg  26083  elntg  26084  edgfiedgval  26122
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