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Theorem tfrcllembacc 6025
 Description: Lemma for tfrcl 6034. Each element of 𝐵 is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f 𝐹 = recs(𝐺)
tfrcl.g (𝜑 → Fun 𝐺)
tfrcl.x (𝜑 → Ord 𝑋)
tfrcl.ex ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
tfrcllemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfrcllembacc.3 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
tfrcllembacc.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfrcllembacc.4 (𝜑𝐷𝑋)
tfrcllembacc.5 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
Assertion
Ref Expression
tfrcllembacc (𝜑𝐵𝐴)
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑦,𝑧   𝐷,𝑓,𝑔,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐷(𝑧,𝑤,)   𝑆(𝑧,𝑤,𝑔,)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐺(𝑧,𝑤,𝑔,)   𝑋(𝑦,𝑧,𝑤,𝑔,)

Proof of Theorem tfrcllembacc
StepHypRef Expression
1 tfrcllembacc.3 . 2 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
2 simpr3 947 . . . . . . 7 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
3 tfrcl.f . . . . . . . 8 𝐹 = recs(𝐺)
4 tfrcl.g . . . . . . . . 9 (𝜑 → Fun 𝐺)
54ad2antrr 472 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Fun 𝐺)
6 tfrcl.x . . . . . . . . 9 (𝜑 → Ord 𝑋)
76ad2antrr 472 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝑋)
8 simp1ll 1002 . . . . . . . . 9 ((((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓:𝑥𝑆) → 𝜑)
9 tfrcl.ex . . . . . . . . 9 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
108, 9syld3an1 1216 . . . . . . . 8 ((((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
11 tfrcllemsucfn.1 . . . . . . . 8 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
12 tfrcllembacc.4 . . . . . . . . 9 (𝜑𝐷𝑋)
1312ad2antrr 472 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐷𝑋)
14 simplr 497 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝐷)
15 tfrcllembacc.u . . . . . . . . . 10 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
1615adantlr 461 . . . . . . . . 9 (((𝜑𝑧𝐷) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
1716adantlr 461 . . . . . . . 8 ((((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
18 simpr1 945 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔:𝑧𝑆)
19 simpr2 946 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
203, 5, 7, 10, 11, 13, 14, 17, 18, 19tfrcllemsucaccv 6024 . . . . . . 7 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
212, 20eqeltrd 2159 . . . . . 6 (((𝜑𝑧𝐷) ∧ (𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐴)
2221ex 113 . . . . 5 ((𝜑𝑧𝐷) → ((𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → 𝐴))
2322exlimdv 1742 . . . 4 ((𝜑𝑧𝐷) → (∃𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → 𝐴))
2423rexlimdva 2482 . . 3 (𝜑 → (∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → 𝐴))
2524abssdv 3077 . 2 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝐴)
261, 25syl5eqss 3052 1 (𝜑𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 920   = wceq 1285  ∃wex 1422   ∈ wcel 1434  {cab 2069  ∀wral 2353  ∃wrex 2354   ∪ cun 2980   ⊆ wss 2982  {csn 3416  ⟨cop 3419  ∪ cuni 3621  Ord word 4145  suc csuc 4148   ↾ cres 4393  Fun wfun 4946  ⟶wf 4948  ‘cfv 4952  recscrecs 5974 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-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308 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-v 2612  df-sbc 2825  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-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960 This theorem is referenced by:  tfrcllembfn  6027  tfrcllemubacc  6029
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