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Theorem rdgisucinc 6034
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6108 and omsuc 6116. (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 6033 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5 unass 3130 . . 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 2132 . 2 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7 rdgival 6031 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
81, 2, 3, 7syl3anc 1170 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
98uneq1d 3126 . 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 6026 . . . . 5 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
111, 2, 3, 10syl3anc 1170 . . . 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 5209 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1513, 14sseq12d 3029 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1615spcgv 2686 . . . 4 ((rec(𝐹, 𝐴)‘𝐵) ∈ V → (∀𝑥 𝑥 ⊆ (𝐹𝑥) → (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1711, 12, 16sylc 61 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
18 ssequn1 3143 . . 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 120 . 2 (𝜑 → ((rec(𝐹, 𝐴)‘𝐵) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
206, 9, 193eqtr2d 2120 1 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283   = wceq 1285  wcel 1434  Vcvv 2602  cun 2972  wss 2974   ciun 3686  Oncon0 4126  suc csuc 4128   Fn wfn 4927  cfv 4932  reccrdg 6018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-recs 5954  df-irdg 6019
This theorem is referenced by:  frecrdg  6057
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