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Theorem frecfcllem 6073
Description: Lemma for frecfcl 6074. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)
Hypothesis
Ref Expression
frecfcllem.g 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
Assertion
Ref Expression
frecfcllem ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Distinct variable groups:   𝐴,𝑔,𝑚,𝑥   𝑔,𝐹,𝑚,𝑥   𝑧,𝐹,𝑚,𝑥   𝑆,𝑚,𝑥,𝑧
Allowed substitution hints:   𝐴(𝑧)   𝑆(𝑔)   𝐺(𝑥,𝑧,𝑔,𝑚)

Proof of Theorem frecfcllem
Dummy variables 𝑓 𝑦 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecfcllem.g . . . . . 6 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
2 funmpt 4988 . . . . . . 7 Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
32a1i 9 . . . . . 6 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
4 ordom 4375 . . . . . . 7 Ord ω
54a1i 9 . . . . . 6 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → Ord ω)
6 vex 2613 . . . . . . . 8 𝑓 ∈ V
7 simp2 940 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑦 ∈ ω)
8 simp3 941 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑓:𝑦𝑆)
9 simp1ll 1002 . . . . . . . . . 10 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
10 fveq2 5229 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
1110eleq1d 2151 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑧) ∈ 𝑆 ↔ (𝐹𝑤) ∈ 𝑆))
1211cbvralv 2582 . . . . . . . . . 10 (∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆 ↔ ∀𝑤𝑆 (𝐹𝑤) ∈ 𝑆)
139, 12sylib 120 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑤𝑆 (𝐹𝑤) ∈ 𝑆)
14 simp1lr 1003 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝐴𝑆)
157, 8, 13, 14frecabcl 6068 . . . . . . . 8 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆)
16 dmeq 4583 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
1716eqeq1d 2091 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
18 fveq1 5228 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → (𝑔𝑚) = (𝑓𝑚))
1918fveq2d 5233 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝐹‘(𝑔𝑚)) = (𝐹‘(𝑓𝑚)))
2019eleq2d 2152 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓𝑚))))
2117, 20anbi12d 457 . . . . . . . . . . . 12 (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2221rexbidv 2374 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2316eqeq1d 2091 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
2423anbi1d 453 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
2522, 24orbi12d 740 . . . . . . . . . 10 (𝑔 = 𝑓 → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))))
2625abbidv 2200 . . . . . . . . 9 (𝑔 = 𝑓 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))})
27 eqid 2083 . . . . . . . . 9 (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
2826, 27fvmptg 5300 . . . . . . . 8 ((𝑓 ∈ V ∧ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))})
296, 15, 28sylancr 405 . . . . . . 7 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))})
3029, 15eqeltrd 2159 . . . . . 6 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) ∈ 𝑆)
31 limom 4382 . . . . . . . . . 10 Lim ω
32 limuni 4179 . . . . . . . . . 10 (Lim ω → ω = ω)
3331, 32ax-mp 7 . . . . . . . . 9 ω = ω
3433eleq2i 2149 . . . . . . . 8 (𝑦 ∈ ω ↔ 𝑦 ω)
35 peano2 4364 . . . . . . . 8 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
3634, 35sylbir 133 . . . . . . 7 (𝑦 ω → suc 𝑦 ∈ ω)
3736adantl 271 . . . . . 6 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ω) → suc 𝑦 ∈ ω)
3833eleq2i 2149 . . . . . . . 8 (𝑘 ∈ ω ↔ 𝑘 ω)
3938biimpi 118 . . . . . . 7 (𝑘 ∈ ω → 𝑘 ω)
4039adantl 271 . . . . . 6 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ω)
411, 3, 5, 30, 37, 40tfrcldm 6032 . . . . 5 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom 𝐺)
421, 3, 5, 30, 37, 40tfrcl 6033 . . . . 5 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → (𝐺𝑘) ∈ 𝑆)
4341, 42jca 300 . . . 4 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆))
4443ralrimiva 2439 . . 3 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆))
45 tfrfun 5989 . . . . 5 Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
461funeqi 4972 . . . . 5 (Fun 𝐺 ↔ Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})))
4745, 46mpbir 144 . . . 4 Fun 𝐺
48 ffvresb 5380 . . . 4 (Fun 𝐺 → ((𝐺 ↾ ω):ω⟶𝑆 ↔ ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆)))
4947, 48ax-mp 7 . . 3 ((𝐺 ↾ ω):ω⟶𝑆 ↔ ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆))
5044, 49sylibr 132 . 2 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → (𝐺 ↾ ω):ω⟶𝑆)
51 df-frec 6060 . . . 4 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
521reseq1i 4656 . . . 4 (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
5351, 52eqtr4i 2106 . . 3 frec(𝐹, 𝐴) = (𝐺 ↾ ω)
5453feq1i 5090 . 2 (frec(𝐹, 𝐴):ω⟶𝑆 ↔ (𝐺 ↾ ω):ω⟶𝑆)
5550, 54sylibr 132 1 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2610  c0 3267   cuni 3621  cmpt 3859  Ord word 4145  Lim wlim 4147  suc csuc 4148  ωcom 4359  dom cdm 4391  cres 4393  Fun wfun 4946  wf 4948  cfv 4952  recscrecs 5973  freccfrec 6059
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5974  df-frec 6060
This theorem is referenced by:  frecfcl  6074
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