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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 13575 | . 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
2 | nfcv 2969 | . . . 4 ⊢ Ⅎ𝑥ℕ0 | |
3 | nfcv 2969 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
4 | nfcv 2969 | . . . . 5 ⊢ Ⅎ𝑥(0..^𝑙) | |
5 | nfwrd.1 | . . . . 5 ⊢ Ⅎ𝑥𝑆 | |
6 | 3, 4, 5 | nff 6274 | . . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆 |
7 | 2, 6 | nfrex 3215 | . . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 |
8 | 7 | nfab 2974 | . 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} |
9 | 1, 8 | nfcxfr 2967 | 1 ⊢ Ⅎ𝑥Word 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: {cab 2811 Ⅎwnfc 2956 ∃wrex 3118 ⟶wf 6119 (class class class)co 6905 0cc0 10252 ℕ0cn0 11618 ..^cfzo 12760 Word cword 13574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-fun 6125 df-fn 6126 df-f 6127 df-word 13575 |
This theorem is referenced by: (None) |
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