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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 2157 | . 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 6265 | 1 ⊢ Fun recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 {cab 2143 ∀wral 2435 ∃wrex 2436 Oncon0 4324 ↾ cres 4589 Fun wfun 5165 Fn wfn 5166 ‘cfv 5171 recscrecs 6252 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-res 4599 df-iota 5136 df-fun 5173 df-fn 5174 df-fv 5179 df-recs 6253 |
This theorem is referenced by: tfr1onlembfn 6292 tfr1onlemubacc 6294 tfri1dALT 6299 tfrcllembfn 6305 tfrcllemubacc 6307 tfrcl 6312 frecex 6342 frecfun 6343 frecfcllem 6352 frecsuclem 6354 |
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