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Theorem tfrcllemubacc 6379
Description: Lemma for tfrcl 6384. The union of 𝐵 satisfies the recursion rule. (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
tfrcllemubacc (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
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   𝐵,𝑔,,𝑧   𝑢,𝐵,𝑤   𝐷,,𝑧   𝑢,𝐷,𝑤   𝑤,𝐺   ,𝐺,𝑧   𝑢,𝐺   𝑆,𝑔,,𝑧   𝑧,𝑋   𝑤,𝑔,𝜑,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦,𝑓)   𝑆(𝑤,𝑢)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑢,𝑔,)

Proof of Theorem tfrcllemubacc
Dummy variables 𝑒 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.f . . . . . . . . 9 𝐹 = recs(𝐺)
2 tfrcl.g . . . . . . . . 9 (𝜑 → Fun 𝐺)
3 tfrcl.x . . . . . . . . 9 (𝜑 → Ord 𝑋)
4 tfrcl.ex . . . . . . . . 9 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
5 tfrcllemsucfn.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6 tfrcllembacc.3 . . . . . . . . 9 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7 tfrcllembacc.u . . . . . . . . 9 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfrcllembacc.4 . . . . . . . . 9 (𝜑𝐷𝑋)
9 tfrcllembacc.5 . . . . . . . . 9 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
101, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembfn 6377 . . . . . . . 8 (𝜑 𝐵:𝐷𝑆)
11 fdm 5387 . . . . . . . 8 ( 𝐵:𝐷𝑆 → dom 𝐵 = 𝐷)
1210, 11syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝐷)
131, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembacc 6375 . . . . . . . . . 10 (𝜑𝐵𝐴)
1413unissd 3848 . . . . . . . . 9 (𝜑 𝐵 𝐴)
155, 3tfrcllemssrecs 6372 . . . . . . . . 9 (𝜑 𝐴 ⊆ recs(𝐺))
1614, 15sstrd 3180 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐺))
17 dmss 4841 . . . . . . . 8 ( 𝐵 ⊆ recs(𝐺) → dom 𝐵 ⊆ dom recs(𝐺))
1816, 17syl 14 . . . . . . 7 (𝜑 → dom 𝐵 ⊆ dom recs(𝐺))
1912, 18eqsstrrd 3207 . . . . . 6 (𝜑𝐷 ⊆ dom recs(𝐺))
2019sselda 3170 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom recs(𝐺))
21 eqid 2189 . . . . . 6 {𝑒 ∣ ∃𝑣 ∈ On (𝑒 Fn 𝑣 ∧ ∀𝑡𝑣 (𝑒𝑡) = (𝐺‘(𝑒𝑡)))} = {𝑒 ∣ ∃𝑣 ∈ On (𝑒 Fn 𝑣 ∧ ∀𝑡𝑣 (𝑒𝑡) = (𝐺‘(𝑒𝑡)))}
2221tfrlem9 6339 . . . . 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 6340 . . . . 5 Fun recs(𝐺)
2512eleq2d 2259 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝐷))
2625biimpar 297 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom 𝐵)
27 funssfv 5557 . . . . 5 ((Fun recs(𝐺) ∧ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ∈ dom 𝐵) → (recs(𝐺)‘𝑤) = ( 𝐵𝑤))
2824, 16, 26, 27mp3an2ani 1355 . . . 4 ((𝜑𝑤𝐷) → (recs(𝐺)‘𝑤) = ( 𝐵𝑤))
29 ordelon 4398 . . . . . . . . . 10 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
303, 8, 29syl2anc 411 . . . . . . . . 9 (𝜑𝐷 ∈ On)
31 eloni 4390 . . . . . . . . 9 (𝐷 ∈ On → Ord 𝐷)
3230, 31syl 14 . . . . . . . 8 (𝜑 → Ord 𝐷)
33 ordelss 4394 . . . . . . . 8 ((Ord 𝐷𝑤𝐷) → 𝑤𝐷)
3432, 33sylan 283 . . . . . . 7 ((𝜑𝑤𝐷) → 𝑤𝐷)
3512adantr 276 . . . . . . 7 ((𝜑𝑤𝐷) → dom 𝐵 = 𝐷)
3634, 35sseqtrrd 3209 . . . . . 6 ((𝜑𝑤𝐷) → 𝑤 ⊆ dom 𝐵)
37 fun2ssres 5275 . . . . . 6 ((Fun recs(𝐺) ∧ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ⊆ dom 𝐵) → (recs(𝐺) ↾ 𝑤) = ( 𝐵𝑤))
3824, 16, 36, 37mp3an2ani 1355 . . . . 5 ((𝜑𝑤𝐷) → (recs(𝐺) ↾ 𝑤) = ( 𝐵𝑤))
3938fveq2d 5535 . . . 4 ((𝜑𝑤𝐷) → (𝐺‘(recs(𝐺) ↾ 𝑤)) = (𝐺‘( 𝐵𝑤)))
4023, 28, 393eqtr3d 2230 . . 3 ((𝜑𝑤𝐷) → ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
4140ralrimiva 2563 . 2 (𝜑 → ∀𝑤𝐷 ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
42 fveq2 5531 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
43 reseq2 4917 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
4443fveq2d 5535 . . . 4 (𝑢 = 𝑤 → (𝐺‘( 𝐵𝑢)) = (𝐺‘( 𝐵𝑤)))
4542, 44eqeq12d 2204 . . 3 (𝑢 = 𝑤 → (( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)) ↔ ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤))))
4645cbvralv 2718 . 2 (∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)) ↔ ∀𝑤𝐷 ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
4741, 46sylibr 134 1 (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wral 2468  wrex 2469  cun 3142  wss 3144  {csn 3607  cop 3610   cuni 3824  Ord word 4377  Oncon0 4378  suc csuc 4380  dom cdm 4641  cres 4643  Fun wfun 5226   Fn wfn 5227  wf 5228  cfv 5232  recscrecs 6324
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-recs 6325
This theorem is referenced by:  tfrcllemex  6380
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