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Theorem rdgisucinc 6411
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6490 and omsuc 6498. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses
Ref Expression
rdgisuc1.1 (𝜑𝐹 Fn V)
rdgisuc1.2 (𝜑𝐴𝑉)
rdgisuc1.3 (𝜑𝐵 ∈ On)
rdgisucinc.inc (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
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
StepHypRef Expression
1 rdgisuc1.1 . . . 4 (𝜑𝐹 Fn V)
2 rdgisuc1.2 . . . 4 (𝜑𝐴𝑉)
3 rdgisuc1.3 . . . 4 (𝜑𝐵 ∈ On)
41, 2, 3rdgisuc1 6410 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5 unass 3307 . . 3 ((𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
64, 5eqtr4di 2240 . 2 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7 rdgival 6408 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
81, 2, 3, 7syl3anc 1249 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
98uneq1d 3303 . 2 (𝜑 → ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = ((𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
10 rdgexggg 6403 . . . . 5 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
111, 2, 3, 10syl3anc 1249 . . . 4 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
12 rdgisucinc.inc . . . 4 (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
13 id 19 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → 𝑥 = (rec(𝐹, 𝐴)‘𝐵))
14 fveq2 5534 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1513, 14sseq12d 3201 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1615spcgv 2839 . . . 4 ((rec(𝐹, 𝐴)‘𝐵) ∈ V → (∀𝑥 𝑥 ⊆ (𝐹𝑥) → (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1711, 12, 16sylc 62 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
18 ssequn1 3320 . . 3 ((rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)) ↔ ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1917, 18sylib 122 . 2 (𝜑 → ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
206, 9, 193eqtr2d 2228 1 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142  wss 3144   ciun 3901  Oncon0 4381  suc csuc 4383   Fn wfn 5230  cfv 5235  reccrdg 6395
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-recs 6331  df-irdg 6396
This theorem is referenced by:  frecrdg  6434
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