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Mirrors > Home > ILE Home > Th. List > tfri2 | GIF version |
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 6380). Here we show that the function 𝐹 has the property that for any function 𝐺 satisfying that condition, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Jim Kingdon, 4-May-2019.) |
Ref | Expression |
---|---|
tfri1.1 | ⊢ 𝐹 = recs(𝐺) |
tfri1.2 | ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) |
Ref | Expression |
---|---|
tfri2 | ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfri1.1 | . . . . 5 ⊢ 𝐹 = recs(𝐺) | |
2 | tfri1.2 | . . . . 5 ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) | |
3 | 1, 2 | tfri1 6380 | . . . 4 ⊢ 𝐹 Fn On |
4 | fndm 5327 | . . . 4 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom 𝐹 = On |
6 | 5 | eleq2i 2254 | . 2 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ On) |
7 | 1 | tfr2a 6336 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
8 | 6, 7 | sylbir 135 | 1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 Vcvv 2749 Oncon0 4375 dom cdm 4638 ↾ cres 4640 Fun wfun 5222 Fn wfn 5223 ‘cfv 5228 recscrecs 6319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-recs 6320 |
This theorem is referenced by: tfri3 6382 |
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