Theorem csbwrdg 14582

Description: Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Assertion

Ref	Expression
csbwrdg	⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 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.

Step	Hyp	Ref	Expression
1		df-word 14553	. . 3 ⊢ Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
2	1	csbeq2i 3907	. 2 ⊢ ⦋𝑆 / 𝑥⦌Word 𝑥 = ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3		sbcrex 3875	. . . . 5 ⊢ ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4		sbcfg 6734	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥))
5		csbconstg 3918	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑤 = 𝑤)
6		csbconstg 3918	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌(0..^𝑙) = (0..^𝑙))
7		csbvarg 4434	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑥 = 𝑆)
8	5, 6, 7	feq123d 6725	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → (⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
9	4, 8	bitrd 279	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
10	9	rexbidv 3179	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
11	3, 10	bitrid 283	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
12	11	abbidv 2808	. . 3 ⊢ (𝑆 ∈ 𝑉 → {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13		csbab 4440	. . 3 ⊢ ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14		df-word 14553	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
15	12, 13, 14	3eqtr4g 2802	. 2 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
16	2, 15	eqtrid 2789	1 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  [wsbc 3788  ⦋csb 3899  ⟶wf 6557  (class class class)co 7431  0cc0 11155  ℕ₀cn0 12526  ..^cfzo 13694  Word cword 14552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-word 14553

This theorem is referenced by:  elovmpowrd  14596