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Theorem tfr1onlemubacc 6015
Description: Lemma for tfr1on 6019. The union of 𝐵 satisfies the recursion rule. (Contributed by Jim Kingdon, 15-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
tfr1onlemubacc (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
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   𝐵,𝑔,,𝑤,𝑧   𝑢,𝐵,𝑤   𝐷,,𝑤,𝑧,𝑓,𝑥   𝑢,𝐷   ,𝐺,𝑧,𝑦   𝑢,𝐺,𝑤   𝑔,𝑋,𝑧   𝜑,𝑤   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑦,𝑢)   𝐴(𝑦,𝑤,𝑢)   𝐵(𝑥,𝑦,𝑓)   𝐷(𝑦)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑢,)

Proof of Theorem tfr1onlemubacc
StepHypRef Expression
1 tfr1on.f . . . . . . . . 9 𝐹 = recs(𝐺)
2 tfr1on.g . . . . . . . . 9 (𝜑 → Fun 𝐺)
3 tfr1on.x . . . . . . . . 9 (𝜑 → Ord 𝑋)
4 tfr1on.ex . . . . . . . . 9 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
5 tfr1onlemsucfn.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6 tfr1onlembacc.3 . . . . . . . . 9 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7 tfr1onlembacc.u . . . . . . . . 9 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfr1onlembacc.4 . . . . . . . . 9 (𝜑𝐷𝑋)
9 tfr1onlembacc.5 . . . . . . . . 9 (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
101, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembfn 6013 . . . . . . . 8 (𝜑 𝐵 Fn 𝐷)
11 fndm 5049 . . . . . . . 8 ( 𝐵 Fn 𝐷 → dom 𝐵 = 𝐷)
1210, 11syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝐷)
131, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembacc 6011 . . . . . . . . . 10 (𝜑𝐵𝐴)
1413unissd 3645 . . . . . . . . 9 (𝜑 𝐵 𝐴)
155, 3tfr1onlemssrecs 6008 . . . . . . . . 9 (𝜑 𝐴 ⊆ recs(𝐺))
1614, 15sstrd 3018 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐺))
17 dmss 4582 . . . . . . . 8 ( 𝐵 ⊆ recs(𝐺) → dom 𝐵 ⊆ dom recs(𝐺))
1816, 17syl 14 . . . . . . 7 (𝜑 → dom 𝐵 ⊆ dom recs(𝐺))
1912, 18eqsstr3d 3043 . . . . . 6 (𝜑𝐷 ⊆ dom recs(𝐺))
2019sselda 3008 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom recs(𝐺))
21 eqid 2083 . . . . . 6 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
2221tfrlem9 5988 . . . . 5 (𝑤 ∈ dom recs(𝐺) → (recs(𝐺)‘𝑤) = (𝐺‘(recs(𝐺) ↾ 𝑤)))
2320, 22syl 14 . . . 4 ((𝜑𝑤𝐷) → (recs(𝐺)‘𝑤) = (𝐺‘(recs(𝐺) ↾ 𝑤)))
24 tfrfun 5989 . . . . 5 Fun recs(𝐺)
2512eleq2d 2152 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝐷))
2625biimpar 291 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom 𝐵)
27 funssfv 5251 . . . . 5 ((Fun recs(𝐺) ∧ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ∈ dom 𝐵) → (recs(𝐺)‘𝑤) = ( 𝐵𝑤))
2824, 16, 26, 27mp3an2ani 1276 . . . 4 ((𝜑𝑤𝐷) → (recs(𝐺)‘𝑤) = ( 𝐵𝑤))
29 ordelon 4166 . . . . . . . . . 10 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
303, 8, 29syl2anc 403 . . . . . . . . 9 (𝜑𝐷 ∈ On)
31 eloni 4158 . . . . . . . . 9 (𝐷 ∈ On → Ord 𝐷)
3230, 31syl 14 . . . . . . . 8 (𝜑 → Ord 𝐷)
33 ordelss 4162 . . . . . . . 8 ((Ord 𝐷𝑤𝐷) → 𝑤𝐷)
3432, 33sylan 277 . . . . . . 7 ((𝜑𝑤𝐷) → 𝑤𝐷)
3512adantr 270 . . . . . . 7 ((𝜑𝑤𝐷) → dom 𝐵 = 𝐷)
3634, 35sseqtr4d 3045 . . . . . 6 ((𝜑𝑤𝐷) → 𝑤 ⊆ dom 𝐵)
37 fun2ssres 4993 . . . . . 6 ((Fun recs(𝐺) ∧ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ⊆ dom 𝐵) → (recs(𝐺) ↾ 𝑤) = ( 𝐵𝑤))
3824, 16, 36, 37mp3an2ani 1276 . . . . 5 ((𝜑𝑤𝐷) → (recs(𝐺) ↾ 𝑤) = ( 𝐵𝑤))
3938fveq2d 5233 . . . 4 ((𝜑𝑤𝐷) → (𝐺‘(recs(𝐺) ↾ 𝑤)) = (𝐺‘( 𝐵𝑤)))
4023, 28, 393eqtr3d 2123 . . 3 ((𝜑𝑤𝐷) → ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
4140ralrimiva 2439 . 2 (𝜑 → ∀𝑤𝐷 ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
42 fveq2 5229 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
43 reseq2 4655 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
4443fveq2d 5233 . . . 4 (𝑢 = 𝑤 → (𝐺‘( 𝐵𝑢)) = (𝐺‘( 𝐵𝑤)))
4542, 44eqeq12d 2097 . . 3 (𝑢 = 𝑤 → (( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)) ↔ ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤))))
4645cbvralv 2582 . 2 (∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)) ↔ ∀𝑤𝐷 ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
4741, 46sylibr 132 1 (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
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  Vcvv 2610  cun 2980  wss 2982  {csn 3416  cop 3419   cuni 3621  Ord word 4145  Oncon0 4146  suc csuc 4148  dom cdm 4391  cres 4393  Fun wfun 4946   Fn wfn 4947  cfv 4952  recscrecs 5973
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-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-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  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-fv 4960  df-recs 5974
This theorem is referenced by:  tfr1onlemex  6016
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