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Theorem freccllem 6071
 Description: Lemma for freccl 6072. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
Hypotheses
Ref Expression
freccl.a (𝜑𝐴𝑆)
freccl.cl ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
freccl.b (𝜑𝐵 ∈ ω)
freccllem.g 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
Assertion
Ref Expression
freccllem (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Distinct variable groups:   𝐴,𝑔,𝑚,𝑥   𝑧,𝐴,𝑚,𝑥   𝑥,𝐵   𝑔,𝐹,𝑚,𝑥   𝑧,𝐹   𝑆,𝑚,𝑥,𝑧   𝜑,𝑚,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑔)   𝐵(𝑧,𝑔,𝑚)   𝑆(𝑔)   𝐺(𝑥,𝑧,𝑔,𝑚)

Proof of Theorem freccllem
Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frec 6060 . . . 4 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
2 freccllem.g . . . . 5 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
32reseq1i 4656 . . . 4 (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
41, 3eqtr4i 2106 . . 3 frec(𝐹, 𝐴) = (𝐺 ↾ ω)
54fveq1i 5230 . 2 (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6 freccl.b . . . 4 (𝜑𝐵 ∈ ω)
7 fvres 5250 . . . 4 (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
86, 7syl 14 . . 3 (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺𝐵))
9 funmpt 4988 . . . . 5 Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
109a1i 9 . . . 4 (𝜑 → Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
11 ordom 4375 . . . . 5 Ord ω
1211a1i 9 . . . 4 (𝜑 → Ord ω)
13 vex 2613 . . . . . 6 𝑓 ∈ V
14 simp2 940 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑦 ∈ ω)
15 simp3 941 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝑓:𝑦𝑆)
16 freccl.cl . . . . . . . . 9 ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)
1716ralrimiva 2439 . . . . . . . 8 (𝜑 → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
18173ad2ant1 960 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆)
19 freccl.a . . . . . . . 8 (𝜑𝐴𝑆)
20193ad2ant1 960 . . . . . . 7 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → 𝐴𝑆)
2114, 15, 18, 20frecabcl 6068 . . . . . 6 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆)
22 dmeq 4583 . . . . . . . . . . . 12 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
2322eqeq1d 2091 . . . . . . . . . . 11 (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24 fveq1 5228 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝑔𝑚) = (𝑓𝑚))
2524fveq2d 5233 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐹‘(𝑔𝑚)) = (𝐹‘(𝑓𝑚)))
2625eleq2d 2152 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓𝑚))))
2723, 26anbi12d 457 . . . . . . . . . 10 (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2827rexbidv 2374 . . . . . . . . 9 (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
2922eqeq1d 2091 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
3029anbi1d 453 . . . . . . . . 9 (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
3128, 30orbi12d 740 . . . . . . . 8 (𝑔 = 𝑓 → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))))
3231abbidv 2200 . . . . . . 7 (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))})
33 eqid 2083 . . . . . . 7 (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))})
3432, 33fvmptg 5300 . . . . . 6 ((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))})
3513, 21, 34sylancr 405 . . . . 5 ((𝜑𝑦 ∈ 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))})
3635, 21eqeltrd 2159 . . . 4 ((𝜑𝑦 ∈ ω ∧ 𝑓:𝑦𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘𝑓) ∈ 𝑆)
37 limom 4382 . . . . . . 7 Lim ω
38 limuni 4179 . . . . . . 7 (Lim ω → ω = ω)
3937, 38ax-mp 7 . . . . . 6 ω = ω
4039eleq2i 2149 . . . . 5 (𝑦 ∈ ω ↔ 𝑦 ω)
41 peano2 4364 . . . . . 6 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
4241adantl 271 . . . . 5 ((𝜑𝑦 ∈ ω) → suc 𝑦 ∈ ω)
4340, 42sylan2br 282 . . . 4 ((𝜑𝑦 ω) → suc 𝑦 ∈ ω)
446, 39syl6eleq 2175 . . . 4 (𝜑𝐵 ω)
452, 10, 12, 36, 43, 44tfrcl 6033 . . 3 (𝜑 → (𝐺𝐵) ∈ 𝑆)
468, 45eqeltrd 2159 . 2 (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
475, 46syl5eqel 2169 1 (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∨ 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:  freccl  6072
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