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Theorem tfr1onlembacc 6357
Description: Lemma for tfr1on 6365. Each element of 𝐵 is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlembacc.3 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
tfr1onlembacc.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfr1onlembacc.4 (𝜑𝐷𝑋)
tfr1onlembacc.5 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
Assertion
Ref Expression
tfr1onlembacc (𝜑𝐵𝐴)
Distinct variable groups:   𝐴,𝑓,𝑔,,𝑥,𝑧   𝐷,𝑓,𝑔,𝑥   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,,𝑥,𝑧   𝑦,𝑔,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑤)   𝐴(𝑦,𝑤)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐷(𝑦,𝑧,𝑤,)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,)   𝐺(𝑧,𝑤,𝑔,)   𝑋(𝑦,𝑧,𝑤,𝑔,)

Proof of Theorem tfr1onlembacc
StepHypRef Expression
1 tfr1onlembacc.3 . 2 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
2 simpr3 1006 . . . . . . 7 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
3 tfr1on.f . . . . . . . 8 𝐹 = recs(𝐺)
4 tfr1on.g . . . . . . . . 9 (𝜑 → Fun 𝐺)
54ad2antrr 488 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Fun 𝐺)
6 tfr1on.x . . . . . . . . 9 (𝜑 → Ord 𝑋)
76ad2antrr 488 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → Ord 𝑋)
8 tfr1on.ex . . . . . . . . . 10 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
983adant1r 1232 . . . . . . . . 9 (((𝜑𝑧𝐷) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
1093adant1r 1232 . . . . . . . 8 ((((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
11 tfr1onlemsucfn.1 . . . . . . . 8 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
12 tfr1onlembacc.4 . . . . . . . . 9 (𝜑𝐷𝑋)
1312ad2antrr 488 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐷𝑋)
14 simplr 528 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑧𝐷)
15 tfr1onlembacc.u . . . . . . . . . 10 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
1615adantlr 477 . . . . . . . . 9 (((𝜑𝑧𝐷) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
1716adantlr 477 . . . . . . . 8 ((((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
18 simpr1 1004 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔 Fn 𝑧)
19 simpr2 1005 . . . . . . . 8 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝑔𝐴)
203, 5, 7, 10, 11, 13, 14, 17, 18, 19tfr1onlemsucaccv 6356 . . . . . . 7 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
212, 20eqeltrd 2264 . . . . . 6 (((𝜑𝑧𝐷) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))) → 𝐴)
2221ex 115 . . . . 5 ((𝜑𝑧𝐷) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → 𝐴))
2322exlimdv 1829 . . . 4 ((𝜑𝑧𝐷) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → 𝐴))
2423rexlimdva 2604 . . 3 (𝜑 → (∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → 𝐴))
2524abssdv 3241 . 2 (𝜑 → { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))} ⊆ 𝐴)
261, 25eqsstrid 3213 1 (𝜑𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 979   = wceq 1363  wex 1502  wcel 2158  {cab 2173  wral 2465  wrex 2466  Vcvv 2749  cun 3139  wss 3141  {csn 3604  cop 3607   cuni 3821  Ord word 4374  suc csuc 4377  cres 4640  Fun wfun 5222   Fn wfn 5223  cfv 5228  recscrecs 6319
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-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
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-v 2751  df-sbc 2975  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-br 4016  df-opab 4077  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236
This theorem is referenced by:  tfr1onlembfn  6359  tfr1onlemubacc  6361
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