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Theorem rdgisucinc 6250
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6328 and omsuc 6336. (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 6249 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5 unass 3203 . . 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, 5syl6eqr 2168 . 2 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7 rdgival 6247 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
81, 2, 3, 7syl3anc 1201 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
98uneq1d 3199 . 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 6242 . . . . 5 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
111, 2, 3, 10syl3anc 1201 . . . 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 5389 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1513, 14sseq12d 3098 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1615spcgv 2747 . . . 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 3216 . . 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 121 . 2 (𝜑 → ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
206, 9, 193eqtr2d 2156 1 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314   = wceq 1316  wcel 1465  Vcvv 2660  cun 3039  wss 3041   ciun 3783  Oncon0 4255  suc csuc 4257   Fn wfn 5088  cfv 5093  reccrdg 6234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170  df-irdg 6235
This theorem is referenced by:  frecrdg  6273
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