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Theorem freccl 5993
 Description: Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
Hypotheses
Ref Expression
freccl.ex (𝜑 → ∀𝑧(𝐹𝑧) ∈ V)
freccl.a (𝜑𝐴𝑆)
freccl.cl ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
freccl.b (𝜑𝐵 ∈ ω)
Assertion
Ref Expression
freccl (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝑆   𝜑,𝑧
Allowed substitution hint:   𝐵(𝑧)

Proof of Theorem freccl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 freccl.b . 2 (𝜑𝐵 ∈ ω)
2 fveq2 5178 . . . . 5 (𝑥 = 𝐵 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝐵))
32eleq1d 2106 . . . 4 (𝑥 = 𝐵 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆))
43imbi2d 219 . . 3 (𝑥 = 𝐵 → ((𝜑 → (frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆) ↔ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)))
5 fveq2 5178 . . . . 5 (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
65eleq1d 2106 . . . 4 (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘∅) ∈ 𝑆))
7 fveq2 5178 . . . . 5 (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
87eleq1d 2106 . . . 4 (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
9 fveq2 5178 . . . . 5 (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
109eleq1d 2106 . . . 4 (𝑥 = suc 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆))
11 freccl.a . . . . . 6 (𝜑𝐴𝑆)
12 frec0g 5983 . . . . . 6 (𝐴𝑆 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
1311, 12syl 14 . . . . 5 (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
1413, 11eqeltrd 2114 . . . 4 (𝜑 → (frec(𝐹, 𝐴)‘∅) ∈ 𝑆)
15 freccl.ex . . . . . . . . . 10 (𝜑 → ∀𝑧(𝐹𝑧) ∈ V)
16 frecfnom 5986 . . . . . . . . . 10 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑆) → frec(𝐹, 𝐴) Fn ω)
1715, 11, 16syl2anc 391 . . . . . . . . 9 (𝜑 → frec(𝐹, 𝐴) Fn ω)
18 funfvex 5192 . . . . . . . . . 10 ((Fun frec(𝐹, 𝐴) ∧ 𝑦 ∈ dom frec(𝐹, 𝐴)) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
1918funfni 4999 . . . . . . . . 9 ((frec(𝐹, 𝐴) Fn ω ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
2017, 19sylan 267 . . . . . . . 8 ((𝜑𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
21 isset 2561 . . . . . . . 8 ((frec(𝐹, 𝐴)‘𝑦) ∈ V ↔ ∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦))
2220, 21sylib 127 . . . . . . 7 ((𝜑𝑦 ∈ ω) → ∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦))
23 freccl.cl . . . . . . . . . . . . 13 ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
2423ex 108 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑆 → (𝐹𝑧) ∈ 𝑆))
2524adantr 261 . . . . . . . . . . 11 ((𝜑𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → (𝑧𝑆 → (𝐹𝑧) ∈ 𝑆))
26 eleq1 2100 . . . . . . . . . . . 12 (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → (𝑧𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
2726adantl 262 . . . . . . . . . . 11 ((𝜑𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → (𝑧𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
28 fveq2 5178 . . . . . . . . . . . . 13 (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → (𝐹𝑧) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
2928eleq1d 2106 . . . . . . . . . . . 12 (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((𝐹𝑧) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
3029adantl 262 . . . . . . . . . . 11 ((𝜑𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → ((𝐹𝑧) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
3125, 27, 303imtr3d 191 . . . . . . . . . 10 ((𝜑𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
3231ex 108 . . . . . . . . 9 (𝜑 → (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
3332exlimdv 1700 . . . . . . . 8 (𝜑 → (∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
3433adantr 261 . . . . . . 7 ((𝜑𝑦 ∈ ω) → (∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
3522, 34mpd 13 . . . . . 6 ((𝜑𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
3615adantr 261 . . . . . . . 8 ((𝜑𝑦 ∈ ω) → ∀𝑧(𝐹𝑧) ∈ V)
3711adantr 261 . . . . . . . 8 ((𝜑𝑦 ∈ ω) → 𝐴𝑆)
38 simpr 103 . . . . . . . 8 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ ω)
39 frecsuc 5991 . . . . . . . 8 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑆𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4036, 37, 38, 39syl3anc 1135 . . . . . . 7 ((𝜑𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4140eleq1d 2106 . . . . . 6 ((𝜑𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
4235, 41sylibrd 158 . . . . 5 ((𝜑𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆))
4342expcom 109 . . . 4 (𝑦 ∈ ω → (𝜑 → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆)))
446, 8, 10, 14, 43finds2 4324 . . 3 (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆))
454, 44vtoclga 2619 . 2 (𝐵 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆))
461, 45mpcom 32 1 (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  suc csuc 4102  ωcom 4313   Fn wfn 4897  ‘cfv 4902  freccfrec 5977 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978 This theorem is referenced by:  frecuzrdgrrn  9194
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