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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 2737 | . . . 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 8423 | . . 3 ⊢ Fun recs(𝐺) |
| 3 | 1 | tfrlem14 8431 | . . 3 ⊢ dom recs(𝐺) = On |
| 4 | df-fn 6564 | . . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) | |
| 5 | 2, 3, 4 | mpbir2an 711 | . 2 ⊢ recs(𝐺) Fn On |
| 6 | tfr.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
| 7 | 6 | fneq1i 6665 | . 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On) |
| 8 | 5, 7 | mpbir 231 | 1 ⊢ 𝐹 Fn On |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 {cab 2714 ∀wral 3061 ∃wrex 3070 dom cdm 5685 ↾ cres 5687 Oncon0 6384 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 recscrecs 8410 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 |
| This theorem is referenced by: tfr2 8438 tfr3 8439 recsfnon 8443 rdgfnon 8458 dfac8alem 10069 dfac12lem1 10184 dfac12lem2 10185 zorn2lem1 10536 zorn2lem2 10537 zorn2lem4 10539 zorn2lem5 10540 zorn2lem6 10541 zorn2lem7 10542 ttukeylem3 10551 ttukeylem5 10553 ttukeylem6 10554 madeval 27891 newval 27894 madef 27895 dnnumch1 43056 dnnumch3lem 43058 dnnumch3 43059 aomclem6 43071 |
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