Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13850 | . 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
2 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥ℕ0 | |
3 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
4 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥(0..^𝑙) | |
5 | nfwrd.1 | . . . . 5 ⊢ Ⅎ𝑥𝑆 | |
6 | 3, 4, 5 | nff 6503 | . . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆 |
7 | 2, 6 | nfrex 3306 | . . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 |
8 | 7 | nfab 2981 | . 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} |
9 | 1, 8 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑥Word 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: {cab 2796 Ⅎwnfc 2958 ∃wrex 3136 ⟶wf 6344 (class class class)co 7145 0cc0 10525 ℕ0cn0 11885 ..^cfzo 13021 Word cword 13849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-word 13850 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |