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Theorem bj-evalidval 33156
 Description: Closed general form of strndxid 15932. Both sides are equal to (𝐹‘𝐴) by bj-evalid 33153 and bj-evalval 33152 respectively, but bj-evalidval 33156 adds something to bj-evalid 33153 and bj-evalval 33152 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
Assertion
Ref Expression
bj-evalidval ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))

Proof of Theorem bj-evalidval
StepHypRef Expression
1 bj-evalid 33153 . . . 4 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
21fveq2d 6233 . . 3 ((𝑉𝑊𝐴𝑉) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
323adant3 1101 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
4 bj-evalval 33152 . . . 4 (𝐹𝑈 → (Slot 𝐴𝐹) = (𝐹𝐴))
54eqcomd 2657 . . 3 (𝐹𝑈 → (𝐹𝐴) = (Slot 𝐴𝐹))
653ad2ant3 1104 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹𝐴) = (Slot 𝐴𝐹))
73, 6eqtrd 2685 1 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   I cid 5052   ↾ cres 5145  ‘cfv 5926  Slot cslot 15903 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-slot 15908 This theorem is referenced by: (None)
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