Theorem csbwrdg 14498

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 14469	. . 3 ⊢ Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
2	1	csbeq2i 3901	. 2 ⊢ ⦋𝑆 / 𝑥⦌Word 𝑥 = ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
3		sbcrex 3869	. . . . 5 ⊢ ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥)
4		sbcfg 6715	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥))
5		csbconstg 3912	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑤 = 𝑤)
6		csbconstg 3912	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌(0..^𝑙) = (0..^𝑙))
7		csbvarg 4431	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑥 = 𝑆)
8	5, 6, 7	feq123d 6706	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → (⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
9	4, 8	bitrd 278	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆))
10	9	rexbidv 3178	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
11	3, 10	bitrid 282	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
12	11	abbidv 2801	. . 3 ⊢ (𝑆 ∈ 𝑉 → {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
13		csbab 4437	. . 3 ⊢ ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥}
14		df-word 14469	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
15	12, 13, 14	3eqtr4g 2797	. 2 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆)
16	2, 15	eqtrid 2784	1 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070  [wsbc 3777  ⦋csb 3893  ⟶wf 6539  (class class class)co 7411  0cc0 11112  ℕ₀cn0 12476  ..^cfzo 13631  Word cword 14468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-word 14469

This theorem is referenced by:  elovmpowrd  14512