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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 2771 | . . . 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 7630 | . . 3 ⊢ Fun recs(𝐺) |
3 | 1 | tfrlem14 7638 | . . 3 ⊢ dom recs(𝐺) = On |
4 | df-fn 6032 | . . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) | |
5 | 2, 3, 4 | mpbir2an 690 | . 2 ⊢ recs(𝐺) Fn On |
6 | tfr.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
7 | 6 | fneq1i 6123 | . 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On) |
8 | 5, 7 | mpbir 221 | 1 ⊢ 𝐹 Fn On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 {cab 2757 ∀wral 3061 ∃wrex 3062 dom cdm 5249 ↾ cres 5251 Oncon0 5864 Fun wfun 6023 Fn wfn 6024 ‘cfv 6029 recscrecs 7618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-wrecs 7557 df-recs 7619 |
This theorem is referenced by: tfr2 7645 tfr3 7646 recsfnon 7650 rdgfnon 7665 dfac8alem 9050 dfac12lem1 9165 dfac12lem2 9166 zorn2lem1 9518 zorn2lem2 9519 zorn2lem4 9521 zorn2lem5 9522 zorn2lem6 9523 zorn2lem7 9524 ttukeylem3 9533 ttukeylem5 9535 ttukeylem6 9536 madeval 32265 dnnumch1 38133 dnnumch3lem 38135 dnnumch3 38136 aomclem6 38148 |
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