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Theorem strfv2d 15899
Description: Deduction version of strfv 15901. (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 15870 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
4 cnvcnv2 5586 . . . . 5 𝑆 = (𝑆 ↾ V)
54fveq1i 6190 . . . 4 (𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx))
6 fvex 6199 . . . . 5 (𝐸‘ndx) ∈ V
7 fvres 6205 . . . . 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 2643 . . 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 3210 . . . . . . . 8 (𝐶𝑊𝐶 ∈ V)
1412, 13syl 17 . . . . . . 7 (𝜑𝐶 ∈ V)
15 opelxpi 5146 . . . . . . 7 (((𝐸‘ndx) ∈ V ∧ 𝐶 ∈ V) → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
166, 14, 15sylancr 695 . . . . . 6 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (V × V))
1711, 16elind 3796 . . . . 5 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ (𝑆 ∩ (V × V)))
18 cnvcnv 5584 . . . . 5 𝑆 = (𝑆 ∩ (V × V))
1917, 18syl6eleqr 2711 . . . 4 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
20 funopfv 6233 . . . 4 (Fun 𝑆 → (⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶))
2110, 19, 20sylc 65 . . 3 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
229, 21syl5eqr 2669 . 2 (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶)
233, 22eqtr2d 2656 1 (𝜑𝐶 = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  Vcvv 3198  cin 3571  cop 4181   × cxp 5110  ccnv 5111  cres 5114  Fun wfun 5880  cfv 5886  ndxcnx 15848  Slot cslot 15850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-res 5124  df-iota 5849  df-fun 5888  df-fv 5894  df-slot 15855
This theorem is referenced by:  strfv2  15900  opelstrbas  15972  eengbas  25855  ebtwntg  25856  ecgrtg  25857  elntg  25858  edgfiedgval  25896
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