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Theorem bj-evalidval 34454
Description: Closed general form of strndxid 16501. Both sides are equal to (𝐹𝐴) by bj-evalid 34452 and bj-evalval 34451 respectively, but bj-evalidval 34454 adds something to bj-evalid 34452 and bj-evalval 34451 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 34452 . . . 4 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
21fveq2d 6656 . . 3 ((𝑉𝑊𝐴𝑉) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
323adant3 1129 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
4 bj-evalval 34451 . . . 4 (𝐹𝑈 → (Slot 𝐴𝐹) = (𝐹𝐴))
54eqcomd 2828 . . 3 (𝐹𝑈 → (𝐹𝐴) = (Slot 𝐴𝐹))
653ad2ant3 1132 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹𝐴) = (Slot 𝐴𝐹))
73, 6eqtrd 2857 1 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114   I cid 5436  cres 5534  cfv 6334  Slot cslot 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-res 5544  df-iota 6293  df-fun 6336  df-fv 6342  df-slot 16478
This theorem is referenced by: (None)
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