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Theorem csbwrdg 14494
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 14465 . . 3 Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
21csbeq2i 3902 . 2 𝑆 / 𝑥Word 𝑥 = 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3 sbcrex 3870 . . . . 5 ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4 sbcfg 6716 . . . . . . 7 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥))
5 csbconstg 3913 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑤 = 𝑤)
6 csbconstg 3913 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥(0..^𝑙) = (0..^𝑙))
7 csbvarg 4432 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑥 = 𝑆)
85, 6, 7feq123d 6707 . . . . . . 7 (𝑆𝑉 → (𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥𝑤:(0..^𝑙)⟶𝑆))
94, 8bitrd 279 . . . . . 6 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑤:(0..^𝑙)⟶𝑆))
109rexbidv 3179 . . . . 5 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
113, 10bitrid 283 . . . 4 (𝑆𝑉 → ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
1211abbidv 2802 . . 3 (𝑆𝑉 → {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13 csbab 4438 . . 3 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14 df-word 14465 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
1512, 13, 143eqtr4g 2798 . 2 (𝑆𝑉𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
162, 15eqtrid 2785 1 (𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  [wsbc 3778  csb 3894  wf 6540  (class class class)co 7409  0cc0 11110  0cn0 12472  ..^cfzo 13627  Word cword 14464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548  df-word 14465
This theorem is referenced by:  elovmpowrd  14508
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