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Mirrors > Home > ILE Home > Th. List > rdgisuc1 | GIF version |
Description: One way of describing the
value of the recursive definition generator at
a successor. There is no condition on the characteristic function 𝐹
other than 𝐹 Fn V. Given that, the resulting
expression
encompasses both the expected successor term
(𝐹‘(rec(𝐹, 𝐴)‘𝐵)) but also terms that correspond to
the initial value 𝐴 and to limit ordinals
∪ 𝑥 ∈ 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)).
If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6282. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Ref | Expression |
---|---|
rdgisuc1.1 | ⊢ (𝜑 → 𝐹 Fn V) |
rdgisuc1.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
rdgisuc1.3 | ⊢ (𝜑 → 𝐵 ∈ On) |
Ref | Expression |
---|---|
rdgisuc1 | ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgisuc1.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn V) | |
2 | rdgisuc1.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | rdgisuc1.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ On) | |
4 | suceloni 4417 | . . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ (𝜑 → suc 𝐵 ∈ On) |
6 | rdgival 6279 | . . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ suc 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)))) | |
7 | 1, 2, 5, 6 | syl3anc 1216 | . 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)))) |
8 | df-suc 4293 | . . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
9 | iuneq1 3826 | . . . . . . 7 ⊢ (suc 𝐵 = (𝐵 ∪ {𝐵}) → ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥))) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) |
11 | iunxun 3892 | . . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) | |
12 | 10, 11 | eqtri 2160 | . . . . 5 ⊢ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) |
13 | fveq2 5421 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝐵)) | |
14 | 13 | fveq2d 5425 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
15 | 14 | iunxsng 3888 | . . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
16 | 15 | uneq2d 3230 | . . . . 5 ⊢ (𝐵 ∈ On → (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))) |
17 | 12, 16 | syl5eq 2184 | . . . 4 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))) |
18 | 17 | uneq2d 3230 | . . 3 ⊢ (𝐵 ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))) |
19 | 3, 18 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))) |
20 | 7, 19 | eqtrd 2172 | 1 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {csn 3527 ∪ ciun 3813 Oncon0 4285 suc csuc 4287 Fn wfn 5118 ‘cfv 5123 reccrdg 6266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-recs 6202 df-irdg 6267 |
This theorem is referenced by: rdgisucinc 6282 |
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