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Theorem frecfcllem 6418
Description: Lemma for frecfcl 6419. 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 5266 . . . . . . 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 4618 . . . . . . 7 Ord ω
54a1i 9 . . . . . 6 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → Ord ω)
6 vex 2752 . . . . . . . 8 𝑓 ∈ V
7 simp2 999 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑦 ∈ ω)
8 simp3 1000 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑓:𝑦𝑆)
9 simp1ll 1061 . . . . . . . . . 10 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
10 fveq2 5527 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
1110eleq1d 2256 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑧) ∈ 𝑆 ↔ (𝐹𝑤) ∈ 𝑆))
1211cbvralv 2715 . . . . . . . . . 10 (∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆 ↔ ∀𝑤𝑆 (𝐹𝑤) ∈ 𝑆)
139, 12sylib 122 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑤𝑆 (𝐹𝑤) ∈ 𝑆)
14 simp1lr 1062 . . . . . . . . 9 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝐴𝑆)
157, 8, 13, 14frecabcl 6413 . . . . . . . 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 4839 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
1716eqeq1d 2196 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
18 fveq1 5526 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → (𝑔𝑚) = (𝑓𝑚))
1918fveq2d 5531 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝐹‘(𝑔𝑚)) = (𝐹‘(𝑓𝑚)))
2019eleq2d 2257 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓𝑚))))
2117, 20anbi12d 473 . . . . . . . . . . . 12 (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2221rexbidv 2488 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2316eqeq1d 2196 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
2423anbi1d 465 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
2522, 24orbi12d 794 . . . . . . . . . 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 2305 . . . . . . . . 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 2187 . . . . . . . . 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 5605 . . . . . . . 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 414 . . . . . . 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 2264 . . . . . 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 4625 . . . . . . . . . 10 Lim ω
32 limuni 4408 . . . . . . . . . 10 (Lim ω → ω = ω)
3331, 32ax-mp 5 . . . . . . . . 9 ω = ω
3433eleq2i 2254 . . . . . . . 8 (𝑦 ∈ ω ↔ 𝑦 ω)
35 peano2 4606 . . . . . . . 8 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
3634, 35sylbir 135 . . . . . . 7 (𝑦 ω → suc 𝑦 ∈ ω)
3736adantl 277 . . . . . 6 ((((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ω) → suc 𝑦 ∈ ω)
3833eleq2i 2254 . . . . . . . 8 (𝑘 ∈ ω ↔ 𝑘 ω)
3938biimpi 120 . . . . . . 7 (𝑘 ∈ ω → 𝑘 ω)
4039adantl 277 . . . . . 6 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ω)
411, 3, 5, 30, 37, 40tfrcldm 6377 . . . . 5 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom 𝐺)
421, 3, 5, 30, 37, 40tfrcl 6378 . . . . 5 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → (𝐺𝑘) ∈ 𝑆)
4341, 42jca 306 . . . 4 (((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) ∧ 𝑘 ∈ ω) → (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆))
4443ralrimiva 2560 . . 3 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆))
45 tfrfun 6334 . . . . 5 Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
461funeqi 5249 . . . . 5 (Fun 𝐺 ↔ Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})))
4745, 46mpbir 146 . . . 4 Fun 𝐺
48 ffvresb 5692 . . . 4 (Fun 𝐺 → ((𝐺 ↾ ω):ω⟶𝑆 ↔ ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆)))
4947, 48ax-mp 5 . . 3 ((𝐺 ↾ ω):ω⟶𝑆 ↔ ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺𝑘) ∈ 𝑆))
5044, 49sylibr 134 . 2 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → (𝐺 ↾ ω):ω⟶𝑆)
51 df-frec 6405 . . . 4 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
521reseq1i 4915 . . . 4 (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
5351, 52eqtr4i 2211 . . 3 frec(𝐹, 𝐴) = (𝐺 ↾ ω)
5453feq1i 5370 . 2 (frec(𝐹, 𝐴):ω⟶𝑆 ↔ (𝐺 ↾ ω):ω⟶𝑆)
5550, 54sylibr 134 1 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709  w3a 979   = wceq 1363  wcel 2158  {cab 2173  wral 2465  wrex 2466  Vcvv 2749  c0 3434   cuni 3821  cmpt 4076  Ord word 4374  Lim wlim 4376  suc csuc 4377  ωcom 4601  dom cdm 4638  cres 4640  Fun wfun 5222  wf 5224  cfv 5228  recscrecs 6318  freccfrec 6404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6319  df-frec 6405
This theorem is referenced by:  frecfcl  6419
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