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Theorem csbwrdg 13605
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 13576 . . 3 Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
21csbeq2i 4218 . 2 𝑆 / 𝑥Word 𝑥 = 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3 sbcrex 3739 . . . . 5 ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4 sbcfg 6277 . . . . . . 7 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥))
5 csbconstg 3771 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑤 = 𝑤)
6 csbconstg 3771 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥(0..^𝑙) = (0..^𝑙))
7 csbvarg 4228 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑥 = 𝑆)
85, 6, 7feq123d 6268 . . . . . . 7 (𝑆𝑉 → (𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥𝑤:(0..^𝑙)⟶𝑆))
94, 8bitrd 271 . . . . . 6 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑤:(0..^𝑙)⟶𝑆))
109rexbidv 3263 . . . . 5 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
113, 10syl5bb 275 . . . 4 (𝑆𝑉 → ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
1211abbidv 2947 . . 3 (𝑆𝑉 → {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13 csbab 4234 . . 3 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14 df-word 13576 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
1512, 13, 143eqtr4g 2887 . 2 (𝑆𝑉𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
162, 15syl5eq 2874 1 (𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  {cab 2812  wrex 3119  [wsbc 3663  csb 3758  wf 6120  (class class class)co 6906  0cc0 10253  0cn0 11619  ..^cfzo 12761  Word cword 13575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-fun 6126  df-fn 6127  df-f 6128  df-word 13576
This theorem is referenced by:  elovmpt2wrd  13619
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