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Theorem strssd 16234
Description: Deduction version of strss 16235. (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 3799 . . 3 (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑇)
71, 2, 3, 6strfvd 16229 . 2 (𝜑𝐶 = (𝐸𝑇))
82, 4ssexd 5000 . . 3 (𝜑𝑆 ∈ V)
9 funss 6120 . . . 4 (𝑆𝑇 → (Fun 𝑇 → Fun 𝑆))
104, 3, 9sylc 65 . . 3 (𝜑 → Fun 𝑆)
111, 8, 10, 5strfvd 16229 . 2 (𝜑𝐶 = (𝐸𝑆))
127, 11eqtr3d 2835 1 (𝜑 → (𝐸𝑇) = (𝐸𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3385  wss 3769  cop 4374  Fun wfun 6095  cfv 6101  ndxcnx 16181  Slot cslot 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-slot 16188
This theorem is referenced by:  strss  16235
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