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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 7645 | . . . 4 ⊢ 𝐹 Fn On |
3 | fndm 6130 | . . . 4 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ dom 𝐹 = On |
5 | 4 | eleq2i 2842 | . 2 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ On) |
6 | 1 | tfr2a 7643 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
7 | 5, 6 | sylbir 225 | 1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 dom cdm 5249 ↾ cres 5251 Oncon0 5866 Fn wfn 6026 ‘cfv 6031 recscrecs 7619 |
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 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 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 5823 df-ord 5869 df-on 5870 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-wrecs 7558 df-recs 7620 |
This theorem is referenced by: tfr3 7647 recsval 7652 rdgval 7668 dfac8alem 9051 dfac12lem1 9166 zorn2lem1 9519 ttukeylem3 9534 madeval 32269 |
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