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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(𝐹‘z) ∈ 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-bndl 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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