Theorem rdgisucinc 6501

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

This can be thought of as a generalization of oasuc 6580 and omsuc 6588. (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 6500	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5		unass 3341	. . 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 2260	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7		rdgival 6498	. . . 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 1252	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
9	8	uneq1d 3337	. 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 6493	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
11	1, 2, 3, 10	syl3anc 1252	. . . 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 5603	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹‘𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
15	13, 14	sseq12d 3235	. . . . 5 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
16	15	spcgv 2870	. . . 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 3354	. . 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 2248	1 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4  ∀wal 1373   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ∪ cun 3175   ⊆ wss 3177  ∪ ciun 3944  Oncon0 4431  suc csuc 4433   Fn wfn 5289  ‘cfv 5294  reccrdg 6485

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-recs 6421  df-irdg 6486

This theorem is referenced by:  frecrdg  6524