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Theorem csbwrdg 13541
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 13506 . . 3 Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
21csbeq2i 4137 . 2 𝑆 / 𝑥Word 𝑥 = 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3 sbcrex 3656 . . . . 5 ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4 sbcfg 6205 . . . . . . 7 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥))
5 csbconstg 3688 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑤 = 𝑤)
6 csbconstg 3688 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥(0..^𝑙) = (0..^𝑙))
7 csbvarg 4147 . . . . . . . 8 (𝑆𝑉𝑆 / 𝑥𝑥 = 𝑆)
85, 6, 7feq123d 6196 . . . . . . 7 (𝑆𝑉 → (𝑆 / 𝑥𝑤:𝑆 / 𝑥(0..^𝑙)⟶𝑆 / 𝑥𝑥𝑤:(0..^𝑙)⟶𝑆))
94, 8bitrd 268 . . . . . 6 (𝑆𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥𝑤:(0..^𝑙)⟶𝑆))
109rexbidv 3191 . . . . 5 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
113, 10syl5bb 272 . . . 4 (𝑆𝑉 → ([𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
1211abbidv 2880 . . 3 (𝑆𝑉 → {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13 csbab 4152 . . 3 𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤[𝑆 / 𝑥]𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14 df-word 13506 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
1512, 13, 143eqtr4g 2820 . 2 (𝑆𝑉𝑆 / 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
162, 15syl5eq 2807 1 (𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  {cab 2747  wrex 3052  [wsbc 3577  csb 3675  wf 6046  (class class class)co 6815  0cc0 10149  0cn0 11505  ..^cfzo 12680  Word cword 13498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-fun 6052  df-fn 6053  df-f 6054  df-word 13506
This theorem is referenced by:  elovmpt2wrd  13555
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