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Theorem strssd 15849
 Description: Deduction version of strss 15850. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
strssd.e 𝐸 = Slot (𝐸‘ndx)
strssd.t (𝜑𝑇𝑉)
strssd.f (𝜑 → Fun 𝑇)
strssd.s (𝜑𝑆𝑇)
strssd.n (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
Assertion
Ref Expression
strssd (𝜑 → (𝐸𝑇) = (𝐸𝑆))

Proof of Theorem strssd
StepHypRef Expression
1 strssd.e . . 3 𝐸 = Slot (𝐸‘ndx)
2 strssd.t . . 3 (𝜑𝑇𝑉)
3 strssd.f . . 3 (𝜑 → Fun 𝑇)
4 strssd.s . . . 4 (𝜑𝑆𝑇)
5 strssd.n . . . 4 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
64, 5sseldd 3589 . . 3 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑇)
71, 2, 3, 6strfvd 15844 . 2 (𝜑𝐶 = (𝐸𝑇))
82, 4ssexd 4775 . . 3 (𝜑𝑆 ∈ V)
9 funss 5876 . . . 4 (𝑆𝑇 → (Fun 𝑇 → Fun 𝑆))
104, 3, 9sylc 65 . . 3 (𝜑 → Fun 𝑆)
111, 8, 10, 5strfvd 15844 . 2 (𝜑𝐶 = (𝐸𝑆))
127, 11eqtr3d 2657 1 (𝜑 → (𝐸𝑇) = (𝐸𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  Vcvv 3190   ⊆ wss 3560  ⟨cop 4161  Fun wfun 5851  ‘cfv 5857  ndxcnx 15797  Slot cslot 15799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-slot 15804 This theorem is referenced by:  strss  15850
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