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Theorem revval 14750
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 3492 . 2 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
2 fveq2 6902 . . . . 5 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
32oveq2d 7442 . . . 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 7441 . . . . . 6 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) βˆ’ 1) = ((β™―β€˜π‘Š) βˆ’ 1))
65oveq1d 7441 . . . . 5 (𝑀 = π‘Š β†’ (((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯) = (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))
74, 6fveq12d 6909 . . . 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 5243 . . 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 14749 . . 3 reverse = (𝑀 ∈ V ↦ (π‘₯ ∈ (0..^(β™―β€˜π‘€)) ↦ (π‘€β€˜(((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯))))
10 ovex 7459 . . . 4 (0..^(β™―β€˜π‘Š)) ∈ V
1110mptex 7241 . . 3 (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))) ∈ V
128, 9, 11fvmpt 7010 . 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 1533   ∈ wcel 2098  Vcvv 3473   ↦ cmpt 5235  β€˜cfv 6553  (class class class)co 7426  0cc0 11146  1c1 11147   βˆ’ cmin 11482  ..^cfzo 13667  β™―chash 14329  reversecreverse 14748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-reverse 14749
This theorem is referenced by:  revcl  14751  revlen  14752  revfv  14753  repswrevw  14777  revco  14825
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