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Mirrors > Home > ILE Home > Th. List > tfrlem6 | GIF version |
Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfrlem6 | ⊢ Rel recs(𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reluni 4745 | . . 3 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑔 ∈ 𝐴 Rel 𝑔) | |
2 | tfrlem.1 | . . . . 5 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
3 | 2 | tfrlem4 6307 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔) |
4 | funrel 5228 | . . . 4 ⊢ (Fun 𝑔 → Rel 𝑔) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔) |
6 | 1, 5 | mprgbir 2535 | . 2 ⊢ Rel ∪ 𝐴 |
7 | 2 | recsfval 6309 | . . 3 ⊢ recs(𝐹) = ∪ 𝐴 |
8 | 7 | releqi 4705 | . 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴) |
9 | 6, 8 | mpbir 146 | 1 ⊢ Rel recs(𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 ∃wrex 2456 ∪ cuni 3807 Oncon0 4359 ↾ cres 4624 Rel wrel 4627 Fun wfun 5205 Fn wfn 5206 ‘cfv 5211 recscrecs 6298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-recs 6299 |
This theorem is referenced by: tfrlem7 6311 |
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