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Theorem rdgisuc1 6293
 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 6294. (Contributed by Jim Kingdon, 9-Jun-2019.)
Hypotheses
Ref Expression
rdgisuc1.1 (𝜑𝐹 Fn V)
rdgisuc1.2 (𝜑𝐴𝑉)
rdgisuc1.3 (𝜑𝐵 ∈ On)
Assertion
Ref Expression
rdgisuc1 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (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 (𝜑𝐴𝑉)
3 rdgisuc1.3 . . . 4 (𝜑𝐵 ∈ On)
4 suceloni 4428 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ On)
53, 4syl 14 . . 3 (𝜑 → suc 𝐵 ∈ On)
6 rdgival 6291 . . 3 ((𝐹 Fn V ∧ 𝐴𝑉 ∧ suc 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
71, 2, 5, 6syl3anc 1217 . 2 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
8 df-suc 4304 . . . . . . 7 suc 𝐵 = (𝐵 ∪ {𝐵})
9 iuneq1 3836 . . . . . . 7 (suc 𝐵 = (𝐵 ∪ {𝐵}) → 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥)))
108, 9ax-mp 5 . . . . . 6 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥))
11 iunxun 3902 . . . . . 6 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)))
1210, 11eqtri 2162 . . . . 5 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)))
13 fveq2 5433 . . . . . . . 8 (𝑥 = 𝐵 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝐵))
1413fveq2d 5437 . . . . . . 7 (𝑥 = 𝐵 → (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1514iunxsng 3898 . . . . . 6 (𝐵 ∈ On → 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1615uneq2d 3237 . . . . 5 (𝐵 ∈ On → ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1712, 16syl5eq 2186 . . . 4 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1817uneq2d 3237 . . 3 (𝐵 ∈ On → (𝐴 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))) = (𝐴 ∪ ( 𝑥𝐵 (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 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
207, 19eqtrd 2174 1 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 2112  Vcvv 2691   ∪ cun 3076  {csn 3534  ∪ ciun 3823  Oncon0 4296  suc csuc 4298   Fn wfn 5130  ‘cfv 5135  reccrdg 6278 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-coll 4053  ax-sep 4056  ax-pow 4108  ax-pr 4142  ax-un 4366  ax-setind 4463 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-ral 2423  df-rex 2424  df-reu 2425  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-nul 3371  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-iun 3825  df-br 3940  df-opab 4000  df-mpt 4001  df-tr 4037  df-id 4226  df-iord 4299  df-on 4301  df-suc 4304  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142  df-fv 5143  df-recs 6214  df-irdg 6279 This theorem is referenced by:  rdgisucinc  6294
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