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Theorem tfrcllemubacc 6384
Description: Lemma for tfrcl 6389. 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 6382 . . . . . . . 8 (𝜑 𝐵:𝐷𝑆)
11 fdm 5390 . . . . . . . 8 ( 𝐵:𝐷𝑆 → dom 𝐵 = 𝐷)
1210, 11syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝐷)
131, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembacc 6380 . . . . . . . . . 10 (𝜑𝐵𝐴)
1413unissd 3848 . . . . . . . . 9 (𝜑 𝐵 𝐴)
155, 3tfrcllemssrecs 6377 . . . . . . . . 9 (𝜑 𝐴 ⊆ recs(𝐺))
1614, 15sstrd 3180 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐺))
17 dmss 4844 . . . . . . . 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 6344 . . . . 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 6345 . . . . 5 Fun recs(𝐺)
2512eleq2d 2259 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝐷))
2625biimpar 297 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom 𝐵)
27 funssfv 5560 . . . . 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 4401 . . . . . . . . . 10 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
303, 8, 29syl2anc 411 . . . . . . . . 9 (𝜑𝐷 ∈ On)
31 eloni 4393 . . . . . . . . 9 (𝐷 ∈ On → Ord 𝐷)
3230, 31syl 14 . . . . . . . 8 (𝜑 → Ord 𝐷)
33 ordelss 4397 . . . . . . . 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 5278 . . . . . 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 5538 . . . 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 5534 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
43 reseq2 4920 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
4443fveq2d 5538 . . . 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 4380  Oncon0 4381  suc csuc 4383  dom cdm 4644  cres 4646  Fun wfun 5229   Fn wfn 5230  wf 5231  cfv 5235  recscrecs 6329
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 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
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 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-recs 6330
This theorem is referenced by:  tfrcllemex  6385
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