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Mirrors > Home > MPE Home > Th. List > tfr1 | Structured version Visualization version GIF version |
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class 𝐺, normally a function, and define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
tfr.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr1 | ⊢ 𝐹 Fn On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem7 8008 | . . 3 ⊢ Fun recs(𝐺) |
3 | 1 | tfrlem14 8016 | . . 3 ⊢ dom recs(𝐺) = On |
4 | df-fn 6351 | . . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) | |
5 | 2, 3, 4 | mpbir2an 707 | . 2 ⊢ recs(𝐺) Fn On |
6 | tfr.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
7 | 6 | fneq1i 6443 | . 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On) |
8 | 5, 7 | mpbir 232 | 1 ⊢ 𝐹 Fn On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 {cab 2796 ∀wral 3135 ∃wrex 3136 dom cdm 5548 ↾ cres 5550 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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 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-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 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-mpt 5138 df-tr 5164 df-id 5453 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-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-wrecs 7936 df-recs 7997 |
This theorem is referenced by: tfr2 8023 tfr3 8024 recsfnon 8028 rdgfnon 8043 dfac8alem 9443 dfac12lem1 9557 dfac12lem2 9558 zorn2lem1 9906 zorn2lem2 9907 zorn2lem4 9909 zorn2lem5 9910 zorn2lem6 9911 zorn2lem7 9912 ttukeylem3 9921 ttukeylem5 9923 ttukeylem6 9924 madeval 33186 dnnumch1 39522 dnnumch3lem 39524 dnnumch3 39525 aomclem6 39537 |
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