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Theorem strfvnd 17124
Description: Deduction version of strfvn 17125. (Contributed by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
strfvnd.c 𝐸 = Slot 𝑁
strfvnd.f (𝜑𝑆𝑉)
Assertion
Ref Expression
strfvnd (𝜑 → (𝐸𝑆) = (𝑆𝑁))

Proof of Theorem strfvnd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 strfvnd.f . 2 (𝜑𝑆𝑉)
2 elex 3487 . 2 (𝑆𝑉𝑆 ∈ V)
3 fveq1 6883 . . 3 (𝑥 = 𝑆 → (𝑥𝑁) = (𝑆𝑁))
4 strfvnd.c . . . 4 𝐸 = Slot 𝑁
5 df-slot 17121 . . . 4 Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥𝑁))
64, 5eqtri 2754 . . 3 𝐸 = (𝑥 ∈ V ↦ (𝑥𝑁))
7 fvex 6897 . . 3 (𝑆𝑁) ∈ V
83, 6, 7fvmpt 6991 . 2 (𝑆 ∈ V → (𝐸𝑆) = (𝑆𝑁))
91, 2, 83syl 18 1 (𝜑 → (𝐸𝑆) = (𝑆𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  cmpt 5224  cfv 6536  Slot cslot 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-slot 17121
This theorem is referenced by:  strfvn  17125  strfvss  17126  strndxid  17137  setsidvald  17138  setsidvaldOLD  17139  strfvd  17140  strfv2d  17141  setsid  17147  setsnid  17148  setsnidOLD  17149  estrreslem1  18097  edgfndxid  28754  bj-endbase  36703  bj-endcomp  36704
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