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Mirrors > Home > MPE Home > Th. List > tfrlem6 | Structured version Visualization version 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 5811 | . . 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 8377 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔) |
4 | funrel 6558 | . . . 4 ⊢ (Fun 𝑔 → Rel 𝑔) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔) |
6 | 1, 5 | mprgbir 3062 | . 2 ⊢ Rel ∪ 𝐴 |
7 | 2 | recsfval 8379 | . . 3 ⊢ recs(𝐹) = ∪ 𝐴 |
8 | 7 | releqi 5770 | . 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴) |
9 | 6, 8 | mpbir 230 | 1 ⊢ Rel recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 ∀wral 3055 ∃wrex 3064 ∪ cuni 4902 ↾ cres 5671 Rel wrel 5674 Oncon0 6357 Fun wfun 6530 Fn wfn 6531 ‘cfv 6536 recscrecs 8368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-ov 7407 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 |
This theorem is referenced by: tfrlem7 8381 tfrlem11 8386 tfrlem15 8390 tfrlem16 8391 |
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