![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > tfrfun | GIF version |
Description: Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
Ref | Expression |
---|---|
tfrfun | ⊢ Fun recs(𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2083 | . 2 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem7 5987 | 1 ⊢ Fun recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1285 {cab 2069 ∀wral 2353 ∃wrex 2354 Oncon0 4146 ↾ cres 4393 Fun wfun 4946 Fn wfn 4947 ‘cfv 4952 recscrecs 5974 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-setind 4308 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-iord 4149 df-on 4151 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-res 4403 df-iota 4917 df-fun 4954 df-fn 4955 df-fv 4960 df-recs 5975 |
This theorem is referenced by: tfr1onlembfn 6014 tfr1onlemubacc 6016 tfri1dALT 6021 tfrcllembfn 6027 tfrcllemubacc 6029 tfrcl 6034 frecex 6064 frecfun 6065 frecfcllem 6074 frecsuclem 6076 |
Copyright terms: Public domain | W3C validator |