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Theorem csbwrdg 13894
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 13865 . . 3 Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
21csbeq2i 3874 . 2 𝑆 / 𝑥Word 𝑥 = 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3 sbcrex 3842 . . . . 5 ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4 sbcfg 6501 . . . . . . 7 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥))
5 csbconstg 3885 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑤 = 𝑤)
6 csbconstg 3885 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥(0..^𝑙) = (0..^𝑙))
7 csbvarg 4366 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑥 = 𝑆)
85, 6, 7feq123d 6492 . . . . . . 7 (𝑆𝑉 → (𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥𝑤:(0..^𝑙)⟶𝑆))
94, 8bitrd 282 . . . . . 6 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑤:(0..^𝑙)⟶𝑆))
109rexbidv 3290 . . . . 5 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
113, 10syl5bb 286 . . . 4 (𝑆𝑉 → ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
1211abbidv 2888 . . 3 (𝑆𝑉 → {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13 csbab 4372 . . 3 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14 df-word 13865 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
1512, 13, 143eqtr4g 2884 . 2 (𝑆𝑉𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
162, 15syl5eq 2871 1 (𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {cab 2802  wrex 3134  [wsbc 3758  csb 3866  wf 6340  (class class class)co 7146  0cc0 10531  0cn0 11892  ..^cfzo 13035  Word cword 13864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-fun 6346  df-fn 6347  df-f 6348  df-word 13865
This theorem is referenced by:  elovmpowrd  13908
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