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Mirrors > Home > MPE Home > Th. List > tfr2 | Structured version Visualization version GIF version |
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function 𝐹 has the property that for any function 𝐺 whatsoever, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
tfr.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr2 | ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfr.1 | . . . . 5 ⊢ 𝐹 = recs(𝐺) | |
2 | 1 | tfr1 7776 | . . . 4 ⊢ 𝐹 Fn On |
3 | fndm 6235 | . . . 4 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ dom 𝐹 = On |
5 | 4 | eleq2i 2851 | . 2 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ On) |
6 | 1 | tfr2a 7774 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
7 | 5, 6 | sylbir 227 | 1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 dom cdm 5355 ↾ cres 5357 Oncon0 5976 Fn wfn 6130 ‘cfv 6135 recscrecs 7750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-wrecs 7689 df-recs 7751 |
This theorem is referenced by: tfr3 7778 recsval 7783 rdgval 7799 dfac8alem 9185 dfac12lem1 9300 zorn2lem1 9653 ttukeylem3 9668 madeval 32524 |
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