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Theorem rdgisucinc 6097
Description: Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6172 and omsuc 6180. (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 6096 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
5 unass 3146 . . 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 2135 . 2 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = ((𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
7 rdgival 6094 . . . 4 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
81, 2, 3, 7syl3anc 1172 . . 3 (𝜑 → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
98uneq1d 3142 . 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 6089 . . . . 5 ((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
111, 2, 3, 10syl3anc 1172 . . . 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 5261 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝐹𝑥) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
1513, 14sseq12d 3044 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘𝐵) → (𝑥 ⊆ (𝐹𝑥) ↔ (rec(𝐹, 𝐴)‘𝐵) ⊆ (𝐹‘(rec(𝐹, 𝐴)‘𝐵))))
1615spcgv 2699 . . . 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 3159 . . 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 2123 1 (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285   = wceq 1287  wcel 1436  Vcvv 2615  cun 2986  wss 2988   ciun 3712  Oncon0 4162  suc csuc 4164   Fn wfn 4972  cfv 4977  reccrdg 6081
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3927  ax-sep 3930  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-iun 3714  df-br 3820  df-opab 3874  df-mpt 3875  df-tr 3910  df-id 4092  df-iord 4165  df-on 4167  df-suc 4170  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-f1 4982  df-fo 4983  df-f1o 4984  df-fv 4985  df-recs 6017  df-irdg 6082
This theorem is referenced by:  frecrdg  6120
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