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Mirrors > Home > MPE Home > Th. List > csbwrdg | Structured version Visualization version GIF version |
Description: Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
Ref | Expression |
---|---|
csbwrdg | ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-word 13576 | . . 3 ⊢ Word 𝑥 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} | |
2 | 1 | csbeq2i 4218 | . 2 ⊢ ⦋𝑆 / 𝑥⦌Word 𝑥 = ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} |
3 | sbcrex 3739 | . . . . 5 ⊢ ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥) | |
4 | sbcfg 6277 | . . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥)) | |
5 | csbconstg 3771 | . . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑤 = 𝑤) | |
6 | csbconstg 3771 | . . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌(0..^𝑙) = (0..^𝑙)) | |
7 | csbvarg 4228 | . . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌𝑥 = 𝑆) | |
8 | 5, 6, 7 | feq123d 6268 | . . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → (⦋𝑆 / 𝑥⦌𝑤:⦋𝑆 / 𝑥⦌(0..^𝑙)⟶⦋𝑆 / 𝑥⦌𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆)) |
9 | 4, 8 | bitrd 271 | . . . . . 6 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ 𝑤:(0..^𝑙)⟶𝑆)) |
10 | 9 | rexbidv 3263 | . . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 [𝑆 / 𝑥]𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆)) |
11 | 3, 10 | syl5bb 275 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ([𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥 ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆)) |
12 | 11 | abbidv 2947 | . . 3 ⊢ (𝑆 ∈ 𝑉 → {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}) |
13 | csbab 4234 | . . 3 ⊢ ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = {𝑤 ∣ [𝑆 / 𝑥]∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} | |
14 | df-word 13576 | . . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
15 | 12, 13, 14 | 3eqtr4g 2887 | . 2 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑥} = Word 𝑆) |
16 | 2, 15 | syl5eq 2874 | 1 ⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {cab 2812 ∃wrex 3119 [wsbc 3663 ⦋csb 3758 ⟶wf 6120 (class class class)co 6906 0cc0 10253 ℕ0cn0 11619 ..^cfzo 12761 Word cword 13575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-fun 6126 df-fn 6127 df-f 6128 df-word 13576 |
This theorem is referenced by: elovmpt2wrd 13619 |
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