Theorem rdgisucinc 6376

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

This can be thought of as a generalization of oasuc 6455 and omsuc 6463. (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 6375	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5		unass 3290	. . 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 2226	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7		rdgival 6373	. . . 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 1238	. . 3 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
9	8	uneq1d 3286	. 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 6368	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
11	1, 2, 3, 10	syl3anc 1238	. . . 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 5507	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹‘𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
15	13, 14	sseq12d 3184	. . . . 5 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹‘𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
16	15	spcgv 2822	. . . 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 3303	. . 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 2214	1 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Syntax hints:   → wi 4  ∀wal 1351   = wceq 1353   ∈ wcel 2146  Vcvv 2735   ∪ cun 3125   ⊆ wss 3127  ∪ ciun 3882  Oncon0 4357  suc csuc 4359   Fn wfn 5203  ‘cfv 5208  reccrdg 6360

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-recs 6296  df-irdg 6361

This theorem is referenced by:  frecrdg  6399