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Theorem revval 14655
Description: Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
revval (π‘Š ∈ 𝑉 β†’ (reverseβ€˜π‘Š) = (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))))
Distinct variable group:   π‘₯,π‘Š
Allowed substitution hint:   𝑉(π‘₯)

Proof of Theorem revval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 elex 3466 . 2 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
2 fveq2 6847 . . . . 5 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
32oveq2d 7378 . . . 4 (𝑀 = π‘Š β†’ (0..^(β™―β€˜π‘€)) = (0..^(β™―β€˜π‘Š)))
4 id 22 . . . . 5 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
52oveq1d 7377 . . . . . 6 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) βˆ’ 1) = ((β™―β€˜π‘Š) βˆ’ 1))
65oveq1d 7377 . . . . 5 (𝑀 = π‘Š β†’ (((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯) = (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))
74, 6fveq12d 6854 . . . 4 (𝑀 = π‘Š β†’ (π‘€β€˜(((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯)) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯)))
83, 7mpteq12dv 5201 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ (0..^(β™―β€˜π‘€)) ↦ (π‘€β€˜(((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯))) = (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))))
9 df-reverse 14654 . . 3 reverse = (𝑀 ∈ V ↦ (π‘₯ ∈ (0..^(β™―β€˜π‘€)) ↦ (π‘€β€˜(((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯))))
10 ovex 7395 . . . 4 (0..^(β™―β€˜π‘Š)) ∈ V
1110mptex 7178 . . 3 (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))) ∈ V
128, 9, 11fvmpt 6953 . 2 (π‘Š ∈ V β†’ (reverseβ€˜π‘Š) = (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))))
131, 12syl 17 1 (π‘Š ∈ 𝑉 β†’ (reverseβ€˜π‘Š) = (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3448   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  1c1 11059   βˆ’ cmin 11392  ..^cfzo 13574  β™―chash 14237  reversecreverse 14653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-reverse 14654
This theorem is referenced by:  revcl  14656  revlen  14657  revfv  14658  repswrevw  14682  revco  14730
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