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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 2170 | . 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 6296 | 1 ⊢ Fun recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 {cab 2156 ∀wral 2448 ∃wrex 2449 Oncon0 4348 ↾ cres 4613 Fun wfun 5192 Fn wfn 5193 ‘cfv 5198 recscrecs 6283 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 df-recs 6284 |
This theorem is referenced by: tfr1onlembfn 6323 tfr1onlemubacc 6325 tfri1dALT 6330 tfrcllembfn 6336 tfrcllemubacc 6338 tfrcl 6343 frecex 6373 frecfun 6374 frecfcllem 6383 frecsuclem 6385 |
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