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Theorem frecrdg 5931
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5918 produces the same results as df-irdg 5897 restricted to 𝜔.

Presumably the theorem would also hold if 𝐹 Fn V were changed to z(𝐹z) V. (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 variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecrdg.1 . . . 4 (φ𝐹 Fn V)
2 vex 2554 . . . . . 6 z V
3 funfvex 5135 . . . . . . 7 ((Fun 𝐹 z dom 𝐹) → (𝐹z) V)
43funfni 4942 . . . . . 6 ((𝐹 Fn V z V) → (𝐹z) V)
52, 4mpan2 401 . . . . 5 (𝐹 Fn V → (𝐹z) V)
65alrimiv 1751 . . . 4 (𝐹 Fn V → z(𝐹z) V)
71, 6syl 14 . . 3 (φz(𝐹z) V)
8 frecrdg.2 . . 3 (φA 𝑉)
9 frecfnom 5925 . . 3 ((z(𝐹z) V A 𝑉) → frec(𝐹, A) Fn 𝜔)
107, 8, 9syl2anc 391 . 2 (φ → frec(𝐹, A) Fn 𝜔)
11 rdgifnon2 5907 . . . 4 ((z(𝐹z) V A 𝑉) → rec(𝐹, A) Fn On)
127, 8, 11syl2anc 391 . . 3 (φ → rec(𝐹, A) Fn On)
13 omsson 4278 . . 3 𝜔 ⊆ On
14 fnssres 4955 . . 3 ((rec(𝐹, A) Fn On 𝜔 ⊆ On) → (rec(𝐹, A) ↾ 𝜔) Fn 𝜔)
1512, 13, 14sylancl 392 . 2 (φ → (rec(𝐹, A) ↾ 𝜔) Fn 𝜔)
16 fveq2 5121 . . . . 5 (x = ∅ → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘∅))
17 fveq2 5121 . . . . 5 (x = ∅ → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘∅))
1816, 17eqeq12d 2051 . . . 4 (x = ∅ → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘∅) = ((rec(𝐹, A) ↾ 𝜔)‘∅)))
19 fveq2 5121 . . . . 5 (x = y → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘y))
20 fveq2 5121 . . . . 5 (x = y → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘y))
2119, 20eqeq12d 2051 . . . 4 (x = y → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)))
22 fveq2 5121 . . . . 5 (x = suc y → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘suc y))
23 fveq2 5121 . . . . 5 (x = suc y → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))
2422, 23eqeq12d 2051 . . . 4 (x = suc y → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y)))
25 frec0g 5922 . . . . . 6 (A 𝑉 → (frec(𝐹, A)‘∅) = A)
268, 25syl 14 . . . . 5 (φ → (frec(𝐹, A)‘∅) = A)
27 peano1 4260 . . . . . . 7 𝜔
28 fvres 5141 . . . . . . 7 (∅ 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘∅) = (rec(𝐹, A)‘∅))
2927, 28ax-mp 7 . . . . . 6 ((rec(𝐹, A) ↾ 𝜔)‘∅) = (rec(𝐹, A)‘∅)
30 rdg0g 5915 . . . . . . 7 (A 𝑉 → (rec(𝐹, A)‘∅) = A)
318, 30syl 14 . . . . . 6 (φ → (rec(𝐹, A)‘∅) = A)
3229, 31syl5eq 2081 . . . . 5 (φ → ((rec(𝐹, A) ↾ 𝜔)‘∅) = A)
3326, 32eqtr4d 2072 . . . 4 (φ → (frec(𝐹, A)‘∅) = ((rec(𝐹, A) ↾ 𝜔)‘∅))
34 simpr 103 . . . . . . . . . 10 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y))
35 fvres 5141 . . . . . . . . . . 11 (y 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘y) = (rec(𝐹, A)‘y))
3635ad2antlr 458 . . . . . . . . . 10 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → ((rec(𝐹, A) ↾ 𝜔)‘y) = (rec(𝐹, A)‘y))
3734, 36eqtrd 2069 . . . . . . . . 9 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘y) = (rec(𝐹, A)‘y))
3837fveq2d 5125 . . . . . . . 8 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (𝐹‘(frec(𝐹, A)‘y)) = (𝐹‘(rec(𝐹, A)‘y)))
397, 8jca 290 . . . . . . . . . 10 (φ → (z(𝐹z) V A 𝑉))
40 frecsuc 5930 . . . . . . . . . . 11 ((z(𝐹z) V A 𝑉 y 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
41403expa 1103 . . . . . . . . . 10 (((z(𝐹z) V A 𝑉) y 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
4239, 41sylan 267 . . . . . . . . 9 ((φ y 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
4342adantr 261 . . . . . . . 8 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
441adantr 261 . . . . . . . . . 10 ((φ y 𝜔) → 𝐹 Fn V)
458adantr 261 . . . . . . . . . 10 ((φ y 𝜔) → A 𝑉)
46 simpr 103 . . . . . . . . . . 11 ((φ y 𝜔) → y 𝜔)
47 nnon 4275 . . . . . . . . . . 11 (y 𝜔 → y On)
4846, 47syl 14 . . . . . . . . . 10 ((φ y 𝜔) → y On)
49 frecrdg.inc . . . . . . . . . . 11 (φx x ⊆ (𝐹x))
5049adantr 261 . . . . . . . . . 10 ((φ y 𝜔) → x x ⊆ (𝐹x))
5144, 45, 48, 50rdgisucinc 5912 . . . . . . . . 9 ((φ y 𝜔) → (rec(𝐹, A)‘suc y) = (𝐹‘(rec(𝐹, A)‘y)))
5251adantr 261 . . . . . . . 8 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (rec(𝐹, A)‘suc y) = (𝐹‘(rec(𝐹, A)‘y)))
5338, 43, 523eqtr4d 2079 . . . . . . 7 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = (rec(𝐹, A)‘suc y))
54 peano2 4261 . . . . . . . . 9 (y 𝜔 → suc y 𝜔)
55 fvres 5141 . . . . . . . . 9 (suc y 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
5654, 55syl 14 . . . . . . . 8 (y 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
5756ad2antlr 458 . . . . . . 7 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
5853, 57eqtr4d 2072 . . . . . 6 (((φ y 𝜔) (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))
5958ex 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)))
6059expcom 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))))
6118, 21, 24, 33, 60finds2 4267 . . 3 (x 𝜔 → (φ → (frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x)))
6261impcom 116 . 2 ((φ x 𝜔) → (frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x))
6310, 15, 62eqfnfvd 5211 1 (φ → frec(𝐹, A) = (rec(𝐹, A) ↾ 𝜔))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256  cres 4290   Fn wfn 4840  cfv 4845  reccrdg 5896  freccfrec 5917
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-irdg 5897  df-frec 5918
This theorem is referenced by: (None)
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