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Theorem bj-evalidval 34263
Description: Closed general form of strndxid 16498. Both sides are equal to (𝐹𝐴) by bj-evalid 34261 and bj-evalval 34260 respectively, but bj-evalidval 34263 adds something to bj-evalid 34261 and bj-evalval 34260 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 34261 . . . 4 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
21fveq2d 6667 . . 3 ((𝑉𝑊𝐴𝑉) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
323adant3 1124 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
4 bj-evalval 34260 . . . 4 (𝐹𝑈 → (Slot 𝐴𝐹) = (𝐹𝐴))
54eqcomd 2824 . . 3 (𝐹𝑈 → (𝐹𝐴) = (Slot 𝐴𝐹))
653ad2ant3 1127 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹𝐴) = (Slot 𝐴𝐹))
73, 6eqtrd 2853 1 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105   I cid 5452  cres 5550  cfv 6348  Slot cslot 16470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-slot 16475
This theorem is referenced by: (None)
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