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Theorem frecrdg 5908
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5898 produces the same results as df-irdg 5878 restricted to 𝜔. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses
Ref Expression
frecrdg.1 (φ𝐹 Fn V)
frecrdg.2 (φA 𝑉)
frecrdg.inc (φx x ⊆ (𝐹x))
Assertion
Ref Expression
frecrdg (φ → frec(𝐹, A) = (rec(𝐹, A) ↾ 𝜔))
Distinct variable groups:   x,A   x,𝐹   x,𝑉   φ,x

Proof of Theorem frecrdg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 frecrdg.1 . . 3 (φ𝐹 Fn V)
2 frecrdg.2 . . 3 (φA 𝑉)
3 frecfnom 5901 . . 3 ((𝐹 Fn V A 𝑉) → frec(𝐹, A) Fn 𝜔)
41, 2, 3syl2anc 393 . 2 (φ → frec(𝐹, A) Fn 𝜔)
5 rdgifnon 5887 . . . 4 ((𝐹 Fn V A 𝑉) → rec(𝐹, A) Fn On)
61, 2, 5syl2anc 393 . . 3 (φ → rec(𝐹, A) Fn On)
7 omsson 4262 . . 3 𝜔 ⊆ On
8 fnssres 4938 . . 3 ((rec(𝐹, A) Fn On 𝜔 ⊆ On) → (rec(𝐹, A) ↾ 𝜔) Fn 𝜔)
96, 7, 8sylancl 394 . 2 (φ → (rec(𝐹, A) ↾ 𝜔) Fn 𝜔)
10 fveq2 5103 . . . . 5 (x = ∅ → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘∅))
11 fveq2 5103 . . . . 5 (x = ∅ → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘∅))
1210, 11eqeq12d 2036 . . . 4 (x = ∅ → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘∅) = ((rec(𝐹, A) ↾ 𝜔)‘∅)))
13 fveq2 5103 . . . . 5 (x = y → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘y))
14 fveq2 5103 . . . . 5 (x = y → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘y))
1513, 14eqeq12d 2036 . . . 4 (x = y → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)))
16 fveq2 5103 . . . . 5 (x = suc y → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘suc y))
17 fveq2 5103 . . . . 5 (x = suc y → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))
1816, 17eqeq12d 2036 . . . 4 (x = suc y → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y)))
19 frec0g 5902 . . . . . 6 ((𝐹 Fn V A 𝑉) → (frec(𝐹, A)‘∅) = A)
201, 2, 19syl2anc 393 . . . . 5 (φ → (frec(𝐹, A)‘∅) = A)
21 peano1 4244 . . . . . . 7 𝜔
22 fvres 5123 . . . . . . 7 (∅ 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘∅) = (rec(𝐹, A)‘∅))
2321, 22ax-mp 7 . . . . . 6 ((rec(𝐹, A) ↾ 𝜔)‘∅) = (rec(𝐹, A)‘∅)
24 rdgi0g 5886 . . . . . . 7 ((𝐹 Fn V A 𝑉) → (rec(𝐹, A)‘∅) = A)
251, 2, 24syl2anc 393 . . . . . 6 (φ → (rec(𝐹, A)‘∅) = A)
2623, 25syl5eq 2066 . . . . 5 (φ → ((rec(𝐹, A) ↾ 𝜔)‘∅) = A)
2720, 26eqtr4d 2057 . . . 4 (φ → (frec(𝐹, A)‘∅) = ((rec(𝐹, A) ↾ 𝜔)‘∅))
28 ax-ia2 100 . . . . . . . . . 10 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y))
29 fvres 5123 . . . . . . . . . . 11 (y 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘y) = (rec(𝐹, A)‘y))
3029ad2antlr 462 . . . . . . . . . 10 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → ((rec(𝐹, A) ↾ 𝜔)‘y) = (rec(𝐹, A)‘y))
3128, 30eqtrd 2054 . . . . . . . . 9 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘y) = (rec(𝐹, A)‘y))
3231fveq2d 5107 . . . . . . . 8 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (𝐹‘(frec(𝐹, A)‘y)) = (𝐹‘(rec(𝐹, A)‘y)))
331, 2jca 290 . . . . . . . . . 10 (φ → (𝐹 Fn V A 𝑉))
34 frecsuc 5907 . . . . . . . . . . 11 ((𝐹 Fn V A 𝑉 y 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
35343expa 1090 . . . . . . . . . 10 (((𝐹 Fn V A 𝑉) y 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
3633, 35sylan 267 . . . . . . . . 9 ((φ y 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
3736adantr 261 . . . . . . . 8 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
381adantr 261 . . . . . . . . . 10 ((φ y 𝜔) → 𝐹 Fn V)
392adantr 261 . . . . . . . . . 10 ((φ y 𝜔) → A 𝑉)
40 ax-ia2 100 . . . . . . . . . . 11 ((φ y 𝜔) → y 𝜔)
41 nnon 4259 . . . . . . . . . . 11 (y 𝜔 → y On)
4240, 41syl 14 . . . . . . . . . 10 ((φ y 𝜔) → y On)
43 frecrdg.inc . . . . . . . . . . 11 (φx x ⊆ (𝐹x))
4443adantr 261 . . . . . . . . . 10 ((φ y 𝜔) → x x ⊆ (𝐹x))
4538, 39, 42, 44rdgisucinc 5892 . . . . . . . . 9 ((φ y 𝜔) → (rec(𝐹, A)‘suc y) = (𝐹‘(rec(𝐹, A)‘y)))
4645adantr 261 . . . . . . . 8 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (rec(𝐹, A)‘suc y) = (𝐹‘(rec(𝐹, A)‘y)))
4732, 37, 463eqtr4d 2064 . . . . . . 7 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = (rec(𝐹, A)‘suc y))
48 peano2 4245 . . . . . . . . 9 (y 𝜔 → suc y 𝜔)
49 fvres 5123 . . . . . . . . 9 (suc y 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
5048, 49syl 14 . . . . . . . 8 (y 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
5150ad2antlr 462 . . . . . . 7 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
5247, 51eqtr4d 2057 . . . . . 6 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))
5352ex 108 . . . . 5 ((φ y 𝜔) → ((frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y)))
5453expcom 109 . . . 4 (y 𝜔 → (φ → ((frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))))
5512, 15, 18, 27, 54finds2 4251 . . 3 (x 𝜔 → (φ → (frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x)))
5655impcom 116 . 2 ((φ x 𝜔) → (frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x))
574, 9, 56eqfnfvd 5193 1 (φ → frec(𝐹, A) = (rec(𝐹, A) ↾ 𝜔))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228   wcel 1374  Vcvv 2535  wss 2894  c0 3201  Oncon0 4049  suc csuc 4051  𝜔com 4240  cres 4274   Fn wfn 4824  cfv 4829  reccrdg 5877  freccfrec 5897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842  df-irdg 5878  df-frec 5898
This theorem is referenced by: (None)
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