Theorem rdgisucinc 6440

Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6519 and omsuc 6527. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses

Ref	Expression
rdgisuc1.1	⊢ (𝜑 → 𝐹 Fn V)
rdgisuc1.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
rdgisuc1.3	⊢ (𝜑 → 𝐵 ∈ On)
rdgisucinc.inc	⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))

Assertion

Ref	Expression
rdgisucinc	⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rdgisucinc

Step	Hyp	Ref	Expression
1		rdgisuc1.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn V)
2		rdgisuc1.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		rdgisuc1.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
4	1, 2, 3	rdgisuc1 6439	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5		unass 3317	. . 3 ⊢ ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
6	4, 5	eqtr4di 2244	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7		rdgival 6437	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
8	1, 2, 3, 7	syl3anc 1249	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
9	8	uneq1d 3313	. 2 ⊢ (𝜑 → ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
10		rdgexggg 6432	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
11	1, 2, 3, 10	syl3anc 1249	. . . 4 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
12		rdgisucinc.inc	. . . 4 ⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))
13		id 19	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → 𝑥 = (rec(𝐹, 𝐴)‘𝐵))
14		fveq2 5555	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹‘𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
15	13, 14	sseq12d 3211	. . . . 5 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
16	15	spcgv 2848	. . . 4 ⊢ ((rec(𝐹, 𝐴)‘𝐵) ∈ V → (∀𝑥 𝑥 ⊆ (𝐹‘𝑥) → (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
17	11, 12, 16	sylc 62	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
18		ssequn1 3330	. . 3 ⊢ ((rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)) ↔ ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
19	17, 18	sylib 122	. 2 ⊢ (𝜑 → ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
20	6, 9, 19	3eqtr2d 2232	1 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∪ cun 3152   ⊆ wss 3154  ∪ ciun 3913  Oncon0 4395  suc csuc 4397   Fn wfn 5250  ‘cfv 5255  reccrdg 6424

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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6360  df-irdg 6425

This theorem is referenced by:  frecrdg  6463