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Mirrors > Home > MPE Home > Th. List > nfwrd | Structured version Visualization version GIF version |
Description: Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
nfwrd.1 | ⊢ Ⅎ𝑥𝑆 |
Ref | Expression |
---|---|
nfwrd | ⊢ Ⅎ𝑥Word 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-word 14035 | . 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
2 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑥ℕ0 | |
3 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
4 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑥(0..^𝑙) | |
5 | nfwrd.1 | . . . . 5 ⊢ Ⅎ𝑥𝑆 | |
6 | 3, 4, 5 | nff 6519 | . . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆 |
7 | 2, 6 | nfrex 3218 | . . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 |
8 | 7 | nfab 2903 | . 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} |
9 | 1, 8 | nfcxfr 2895 | 1 ⊢ Ⅎ𝑥Word 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: {cab 2714 Ⅎwnfc 2877 ∃wrex 3052 ⟶wf 6354 (class class class)co 7191 0cc0 10694 ℕ0cn0 12055 ..^cfzo 13203 Word cword 14034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 df-f 6362 df-word 14035 |
This theorem is referenced by: (None) |
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