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Theorem csbwrdg 10933
Description: Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
csbwrdg (𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
Distinct variable groups:   𝑥,𝑆   𝑥,𝑉

Proof of Theorem csbwrdg
Dummy variables 𝑙 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-word 10905 . . 3 Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
21csbeq2i 3107 . 2 𝑆 / 𝑥Word 𝑥 = 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3 sbcrex 3065 . . . . 5 ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4 sbcfg 5394 . . . . . . 7 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥))
5 csbconstg 3094 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑤 = 𝑤)
6 csbconstg 3094 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥(0..^𝑙) = (0..^𝑙))
7 csbvarg 3108 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑥 = 𝑆)
85, 6, 7feq123d 5386 . . . . . . 7 (𝑆𝑉 → (𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥𝑤:(0..^𝑙)⟶𝑆))
94, 8bitrd 188 . . . . . 6 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑤:(0..^𝑙)⟶𝑆))
109rexbidv 2495 . . . . 5 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
113, 10bitrid 192 . . . 4 (𝑆𝑉 → ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
1211abbidv 2311 . . 3 (𝑆𝑉 → {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13 csbabg 3142 . . 3 (𝑆𝑉𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥})
14 df-word 10905 . . . 4 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
1514a1i 9 . . 3 (𝑆𝑉 → Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
1612, 13, 153eqtr4d 2236 . 2 (𝑆𝑉𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
172, 16eqtrid 2238 1 (𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2164  {cab 2179  wrex 2473  [wsbc 2985  csb 3080  wf 5242  (class class class)co 5910  0cc0 7862  0cn0 9230  ..^cfzo 10198  Word cword 10904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4322  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-fun 5248  df-fn 5249  df-f 5250  df-word 10905
This theorem is referenced by:  elovmpowrd  10945
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