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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 8296 | . 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 2903 | . . . 4 ⊢ Ⅎ𝑥2nd | |
6 | 4, 5 | nfco 5865 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd ) |
7 | 2, 3, 6 | nffrecs 8267 | . 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) |
8 | 1, 7 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2883 ∘ ccom 5680 2nd c2nd 7973 frecscfrecs 8264 wrecscwrecs 8295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-iota 6495 df-fv 6551 df-ov 7411 df-frecs 8265 df-wrecs 8296 |
This theorem is referenced by: nfrecs 8374 |
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