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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 2728 | . . . 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 8398 | . . 3 ⊢ Fun recs(𝐺) |
3 | 1 | tfrlem14 8406 | . . 3 ⊢ dom recs(𝐺) = On |
4 | df-fn 6546 | . . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) | |
5 | 2, 3, 4 | mpbir2an 710 | . 2 ⊢ recs(𝐺) Fn On |
6 | tfr.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
7 | 6 | fneq1i 6646 | . 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On) |
8 | 5, 7 | mpbir 230 | 1 ⊢ 𝐹 Fn On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 {cab 2705 ∀wral 3057 ∃wrex 3066 dom cdm 5673 ↾ cres 5675 Oncon0 6364 Fun wfun 6537 Fn wfn 6538 ‘cfv 6543 recscrecs 8385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 |
This theorem is referenced by: tfr2 8413 tfr3 8414 recsfnon 8418 rdgfnon 8433 dfac8alem 10047 dfac12lem1 10161 dfac12lem2 10162 zorn2lem1 10514 zorn2lem2 10515 zorn2lem4 10517 zorn2lem5 10518 zorn2lem6 10519 zorn2lem7 10520 ttukeylem3 10529 ttukeylem5 10531 ttukeylem6 10532 madeval 27773 newval 27776 madef 27777 dnnumch1 42459 dnnumch3lem 42461 dnnumch3 42462 aomclem6 42474 |
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