Theorem csbwrdg 14504

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 14474	. . 3 ⊢ Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
2	1	csbeq2i 3846	. 2 ⊢ ⦋𝑆 / 𝑥⦌Word 𝑥 = ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3		sbcrex 3814	. . . . 5 ⊢ ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4		sbcfg 6660	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥))
5		csbconstg 3857	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑤 = 𝑤)
6		csbconstg 3857	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌(0..^𝑙) = (0..^𝑙))
7		csbvarg 4369	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑥 = 𝑆)
8	5, 6, 7	feq123d 6651	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → (⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
9	4, 8	bitrd 280	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
10	9	rexbidv 3164	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
11	3, 10	bitrid 284	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
12	11	abbidv 2806	. . 3 ⊢ (𝑆 ∈ 𝑉 → {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13		csbab 4375	. . 3 ⊢ ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14		df-word 14474	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
15	12, 13, 14	3eqtr4g 2800	. 2 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
16	2, 15	eqtrid 2787	1 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {cab 2718  ∃wrex 3064  [wsbc 3730  ⦋csb 3838  ⟶wf 6488  (class class class)co 7363  0cc0 11036  ℕ₀cn0 12435  ..^cfzo 13606  Word cword 14473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495  df-f 6496  df-word 14474

This theorem is referenced by:  elovmpowrd  14518