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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 2610 | . . . 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 7366 | . . 3 ⊢ Fun recs(𝐺) |
3 | 1 | tfrlem14 7374 | . . 3 ⊢ dom recs(𝐺) = On |
4 | df-fn 5807 | . . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) | |
5 | 2, 3, 4 | mpbir2an 957 | . 2 ⊢ recs(𝐺) Fn On |
6 | tfr.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
7 | 6 | fneq1i 5899 | . 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On) |
8 | 5, 7 | mpbir 220 | 1 ⊢ 𝐹 Fn On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 {cab 2596 ∀wral 2896 ∃wrex 2897 dom cdm 5038 ↾ cres 5040 Oncon0 5640 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 recscrecs 7354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-wrecs 7294 df-recs 7355 |
This theorem is referenced by: tfr2 7381 tfr3 7382 recsfnon 7386 rdgfnon 7401 dfac8alem 8735 dfac12lem1 8848 dfac12lem2 8849 zorn2lem1 9201 zorn2lem2 9202 zorn2lem4 9204 zorn2lem5 9205 zorn2lem6 9206 zorn2lem7 9207 ttukeylem3 9216 ttukeylem5 9218 ttukeylem6 9219 dnnumch1 36632 dnnumch3lem 36634 dnnumch3 36635 aomclem6 36647 |
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