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| Mirrors > Home > MPE Home > Th. List > nfwrecs | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| nfwrecs.1 | ⊢ Ⅎ𝑥𝑅 | 
| nfwrecs.2 | ⊢ Ⅎ𝑥𝐴 | 
| nfwrecs.3 | ⊢ Ⅎ𝑥𝐹 | 
| Ref | Expression | 
|---|---|
| nfwrecs | ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-wrecs 8337 | . 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) | |
| 2 | nfwrecs.1 | . . 3 ⊢ Ⅎ𝑥𝑅 | |
| 3 | nfwrecs.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 4 | nfwrecs.3 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
| 5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥2nd | |
| 6 | 4, 5 | nfco 5876 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd ) | 
| 7 | 2, 3, 6 | nffrecs 8308 | . 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) | 
| 8 | 1, 7 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: Ⅎwnfc 2890 ∘ ccom 5689 2nd c2nd 8013 frecscfrecs 8305 wrecscwrecs 8336 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-ov 7434 df-frecs 8306 df-wrecs 8337 | 
| This theorem is referenced by: nfrecs 8415 | 
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