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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 8244 | . 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 2904 | . . . 4 ⊢ Ⅎ𝑥2nd | |
6 | 4, 5 | nfco 5822 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd ) |
7 | 2, 3, 6 | nffrecs 8215 | . 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) |
8 | 1, 7 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2884 ∘ ccom 5638 2nd c2nd 7921 frecscfrecs 8212 wrecscwrecs 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fv 6505 df-ov 7361 df-frecs 8213 df-wrecs 8244 |
This theorem is referenced by: nfrecs 8322 |
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