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Theorem bj-evalidval 35354
Description: Closed general form of strndxid 16996. Both sides are equal to (𝐹𝐴) by bj-evalid 35352 and bj-evalval 35351 respectively, but bj-evalidval 35354 adds something to bj-evalid 35352 and bj-evalval 35351 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 35352 . . . 4 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
21fveq2d 6829 . . 3 ((𝑉𝑊𝐴𝑉) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
323adant3 1131 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (𝐹𝐴))
4 bj-evalval 35351 . . . 4 (𝐹𝑈 → (Slot 𝐴𝐹) = (𝐹𝐴))
54eqcomd 2742 . . 3 (𝐹𝑈 → (𝐹𝐴) = (Slot 𝐴𝐹))
653ad2ant3 1134 . 2 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹𝐴) = (Slot 𝐴𝐹))
73, 6eqtrd 2776 1 ((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105   I cid 5517  cres 5622  cfv 6479  Slot cslot 16979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-res 5632  df-iota 6431  df-fun 6481  df-fv 6487  df-slot 16980
This theorem is referenced by: (None)
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