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Theorem revval 14714
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 3487 . 2 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
2 fveq2 6884 . . . . 5 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
32oveq2d 7420 . . . 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 7419 . . . . . 6 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) βˆ’ 1) = ((β™―β€˜π‘Š) βˆ’ 1))
65oveq1d 7419 . . . . 5 (𝑀 = π‘Š β†’ (((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯) = (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))
74, 6fveq12d 6891 . . . 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 5232 . . 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 14713 . . 3 reverse = (𝑀 ∈ V ↦ (π‘₯ ∈ (0..^(β™―β€˜π‘€)) ↦ (π‘€β€˜(((β™―β€˜π‘€) βˆ’ 1) βˆ’ π‘₯))))
10 ovex 7437 . . . 4 (0..^(β™―β€˜π‘Š)) ∈ V
1110mptex 7219 . . 3 (π‘₯ ∈ (0..^(β™―β€˜π‘Š)) ↦ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯))) ∈ V
128, 9, 11fvmpt 6991 . 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 3468   ↦ cmpt 5224  β€˜cfv 6536  (class class class)co 7404  0cc0 11109  1c1 11110   βˆ’ cmin 11445  ..^cfzo 13630  β™―chash 14293  reversecreverse 14712
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-reverse 14713
This theorem is referenced by:  revcl  14715  revlen  14716  revfv  14717  repswrevw  14741  revco  14789
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