Theorem csbwrdg 13189

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 13154	. . 3 ⊢ Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
2	1	csbeq2i 3945	. 2 ⊢ ⦋𝑆 / 𝑥⦌Word 𝑥 = ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3		sbcrex 3481	. . . . 5 ⊢ ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4		sbcfg 5956	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥))
5		csbconstg 3512	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑤 = 𝑤)
6		csbconstg 3512	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌(0..^𝑙) = (0..^𝑙))
7		csbvarg 3955	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑥 = 𝑆)
8	5, 6, 7	feq123d 5947	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → (⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
9	4, 8	bitrd 267	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
10	9	rexbidv 3034	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
11	3, 10	syl5bb 271	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
12	11	abbidv 2728	. . 3 ⊢ (𝑆 ∈ 𝑉 → {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13		csbab 3960	. . 3 ⊢ ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14		df-word 13154	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
15	12, 13, 14	3eqtr4g 2669	. 2 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
16	2, 15	syl5eq 2656	1 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  [wsbc 3402  ⦋csb 3499  ⟶wf 5800  (class class class)co 6549  0cc0 9815  ℕ₀cn0 11169  ..^cfzo 12334  Word cword 13146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-word 13154

This theorem is referenced by:  elovmpt2wrd  13202