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Mirrors > Home > ILE Home > Th. List > rdgfun | GIF version |
Description: The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
rdgfun | ⊢ Fun rec(𝐹, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . 3 ⊢ {𝑓 ∣ ∃𝑦 ∈ On (𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(𝑓 ↾ 𝑧)))} = {𝑓 ∣ ∃𝑦 ∈ On (𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(𝑓 ↾ 𝑧)))} | |
2 | 1 | tfrlem7 6214 | . 2 ⊢ Fun recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) |
3 | df-irdg 6267 | . . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) | |
4 | 3 | funeqi 5144 | . 2 ⊢ (Fun rec(𝐹, 𝐴) ↔ Fun recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))) |
5 | 2, 4 | mpbir 145 | 1 ⊢ Fun rec(𝐹, 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 {cab 2125 ∀wral 2416 ∃wrex 2417 Vcvv 2686 ∪ cun 3069 ∪ ciun 3813 ↦ cmpt 3989 Oncon0 4285 dom cdm 4539 ↾ cres 4541 Fun wfun 5117 Fn wfn 5118 ‘cfv 5123 recscrecs 6201 reccrdg 6266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-recs 6202 df-irdg 6267 |
This theorem is referenced by: rdgivallem 6278 |
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