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Theorem rdgisuc1 5911
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 5912. (Contributed by Jim Kingdon, 9-Jun-2019.)

Hypotheses
Ref Expression
rdgisuc1.1 (φ𝐹 Fn V)
rdgisuc1.2 (φA 𝑉)
rdgisuc1.3 (φB On)
Assertion
Ref Expression
rdgisuc1 (φ → (rec(𝐹, A)‘suc B) = (A ∪ ( x B (𝐹‘(rec(𝐹, A)‘x)) ∪ (𝐹‘(rec(𝐹, A)‘B)))))
Distinct variable groups:   x,𝐹   x,A   x,B   x,𝑉
Allowed substitution hint:   φ(x)

Proof of Theorem rdgisuc1
StepHypRef Expression
1 rdgisuc1.1 . . 3 (φ𝐹 Fn V)
2 rdgisuc1.2 . . 3 (φA 𝑉)
3 rdgisuc1.3 . . . 4 (φB On)
4 suceloni 4193 . . . 4 (B On → suc B On)
53, 4syl 14 . . 3 (φ → suc B On)
6 rdgival 5909 . . 3 ((𝐹 Fn V A 𝑉 suc B On) → (rec(𝐹, A)‘suc B) = (A x suc B(𝐹‘(rec(𝐹, A)‘x))))
71, 2, 5, 6syl3anc 1134 . 2 (φ → (rec(𝐹, A)‘suc B) = (A x suc B(𝐹‘(rec(𝐹, A)‘x))))
8 df-suc 4074 . . . . . . 7 suc B = (B ∪ {B})
9 iuneq1 3661 . . . . . . 7 (suc B = (B ∪ {B}) → x suc B(𝐹‘(rec(𝐹, A)‘x)) = x (B ∪ {B})(𝐹‘(rec(𝐹, A)‘x)))
108, 9ax-mp 7 . . . . . 6 x suc B(𝐹‘(rec(𝐹, A)‘x)) = x (B ∪ {B})(𝐹‘(rec(𝐹, A)‘x))
11 iunxun 3726 . . . . . 6 x (B ∪ {B})(𝐹‘(rec(𝐹, A)‘x)) = ( x B (𝐹‘(rec(𝐹, A)‘x)) ∪ x {B} (𝐹‘(rec(𝐹, A)‘x)))
1210, 11eqtri 2057 . . . . 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 5121 . . . . . . . 8 (x = B → (rec(𝐹, A)‘x) = (rec(𝐹, A)‘B))
1413fveq2d 5125 . . . . . . 7 (x = B → (𝐹‘(rec(𝐹, A)‘x)) = (𝐹‘(rec(𝐹, A)‘B)))
1514iunxsng 3723 . . . . . 6 (B On → x {B} (𝐹‘(rec(𝐹, A)‘x)) = (𝐹‘(rec(𝐹, A)‘B)))
1615uneq2d 3091 . . . . 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))))
1712, 16syl5eq 2081 . . . 4 (B On → x suc B(𝐹‘(rec(𝐹, A)‘x)) = ( x B (𝐹‘(rec(𝐹, A)‘x)) ∪ (𝐹‘(rec(𝐹, A)‘B))))
1817uneq2d 3091 . . 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)))))
193, 18syl 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)))))
207, 19eqtrd 2069 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 1242   wcel 1390  Vcvv 2551  cun 2909  {csn 3367   ciun 3648  Oncon0 4066  suc csuc 4068   Fn wfn 4840  cfv 4845  reccrdg 5896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-irdg 5897
This theorem is referenced by:  rdgisucinc  5912
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