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Mirrors > Home > ILE Home > Th. List > nfwrd | 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 10905 | . 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
2 | nfcv 2336 | . . . 4 ⊢ Ⅎ𝑥ℕ0 | |
3 | nfcv 2336 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
4 | nfcv 2336 | . . . . 5 ⊢ Ⅎ𝑥(0..^𝑙) | |
5 | nfwrd.1 | . . . . 5 ⊢ Ⅎ𝑥𝑆 | |
6 | 3, 4, 5 | nff 5392 | . . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆 |
7 | 2, 6 | nfrexw 2533 | . . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 |
8 | 7 | nfab 2341 | . 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} |
9 | 1, 8 | nfcxfr 2333 | 1 ⊢ Ⅎ𝑥Word 𝑆 |
Colors of variables: wff set class |
Syntax hints: {cab 2179 Ⅎwnfc 2323 ∃wrex 2473 ⟶wf 5242 (class class class)co 5910 0cc0 7862 ℕ0cn0 9230 ..^cfzo 10198 Word cword 10904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-fun 5248 df-fn 5249 df-f 5250 df-word 10905 |
This theorem is referenced by: (None) |
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