Theorem rdgisuc1 5994

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 5995. (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		suceloni 4245	. . . 4 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → suc 𝐵 ∈ On)
6		rdgival 5992	. . 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 1169	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
8		df-suc 4126	. . . . . . 7 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
9		iuneq1 3691	. . . . . . 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 7	. . . . . 6 ⊢ ∪ 𝑥 ∈ suc 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = ∪ 𝑥 ∈ (𝐵 ∪ {𝐵})(𝐹‘(rec(𝐹, 𝐴)‘𝑥))
11		iunxun 3756	. . . . . 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 2101	. . . . 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 5198	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (rec(𝐹, 𝐴)‘𝑥) = (rec(𝐹, 𝐴)‘𝐵))
14	13	fveq2d 5202	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
15	14	iunxsng 3753	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝑥 ∈ {𝐵} (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
16	15	uneq2d 3126	. . . . 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 2125	. . . 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 3126	. . 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 2113	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 1284   ∈ wcel 1433  Vcvv 2601   ∪ cun 2971  {csn 3398  ∪ ciun 3678  Oncon0 4118  suc csuc 4120   Fn wfn 4917  ‘cfv 4922  reccrdg 5979

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  rdgisucinc  5995