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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 5684 | . . 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 8004 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔) |
4 | funrel 6365 | . . . 4 ⊢ (Fun 𝑔 → Rel 𝑔) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔) |
6 | 1, 5 | mprgbir 3150 | . 2 ⊢ Rel ∪ 𝐴 |
7 | 2 | recsfval 8006 | . . 3 ⊢ recs(𝐹) = ∪ 𝐴 |
8 | 7 | releqi 5645 | . 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴) |
9 | 6, 8 | mpbir 232 | 1 ⊢ Rel recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 ∃wrex 3136 ∪ cuni 4830 ↾ cres 5550 Rel wrel 5553 Oncon0 6184 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 recscrecs 7996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-wrecs 7936 df-recs 7997 |
This theorem is referenced by: tfrlem7 8008 tfrlem11 8013 tfrlem15 8017 tfrlem16 8018 |
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