Theorem rdgisuc1 6437

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 6438. (Contributed by Jim Kingdon, 9-Jun-2019.)

Hypotheses

Ref	Expression
rdgisuc1.1	⊢ (𝜑 → 𝐹 Fn V)
rdgisuc1.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rdgisuc1.3	⊢ (𝜑 → 𝐵 ∈ 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

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		onsuc 4533	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → suc 𝐵 ∈ On)
6		rdgival 6435	. . 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 1249	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
8		df-suc 4402	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
9		iuneq1 3925	. . . . . . 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 3992	. . . . . 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 2214	. . . . 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 5554	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝐵))
14	13	fveq2d 5558	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
15	14	iunxsng 3988	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
16	15	uneq2d 3313	. . . . 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	eqtrid 2238	. . . 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 3313	. . 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 2226	1 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∪ cun 3151  {csn 3618  ∪ ciun 3912  Oncon0 4394  suc csuc 4396   Fn wfn 5249  ‘cfv 5254  reccrdg 6422

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358  df-irdg 6423

This theorem is referenced by:  rdgisucinc  6438