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Theorem tfrcllemubacc 6256
Description: Lemma for tfrcl 6261. 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 6254 . . . . . . . 8 (𝜑 𝐵:𝐷𝑆)
11 fdm 5278 . . . . . . . 8 ( 𝐵:𝐷𝑆 → dom 𝐵 = 𝐷)
1210, 11syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝐷)
131, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembacc 6252 . . . . . . . . . 10 (𝜑𝐵𝐴)
1413unissd 3760 . . . . . . . . 9 (𝜑 𝐵 𝐴)
155, 3tfrcllemssrecs 6249 . . . . . . . . 9 (𝜑 𝐴 ⊆ recs(𝐺))
1614, 15sstrd 3107 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐺))
17 dmss 4738 . . . . . . . 8 ( 𝐵 ⊆ recs(𝐺) → dom 𝐵 ⊆ dom recs(𝐺))
1816, 17syl 14 . . . . . . 7 (𝜑 → dom 𝐵 ⊆ dom recs(𝐺))
1912, 18eqsstrrd 3134 . . . . . 6 (𝜑𝐷 ⊆ dom recs(𝐺))
2019sselda 3097 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom recs(𝐺))
21 eqid 2139 . . . . . 6 {𝑒 ∣ ∃𝑣 ∈ On (𝑒 Fn 𝑣 ∧ ∀𝑡𝑣 (𝑒𝑡) = (𝐺‘(𝑒𝑡)))} = {𝑒 ∣ ∃𝑣 ∈ On (𝑒 Fn 𝑣 ∧ ∀𝑡𝑣 (𝑒𝑡) = (𝐺‘(𝑒𝑡)))}
2221tfrlem9 6216 . . . . 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 6217 . . . . 5 Fun recs(𝐺)
2512eleq2d 2209 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝐷))
2625biimpar 295 . . . . 5 ((𝜑𝑤𝐷) → 𝑤 ∈ dom 𝐵)
27 funssfv 5447 . . . . 5 ((Fun recs(𝐺) ∧ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ∈ dom 𝐵) → (recs(𝐺)‘𝑤) = ( 𝐵𝑤))
2824, 16, 26, 27mp3an2ani 1322 . . . 4 ((𝜑𝑤𝐷) → (recs(𝐺)‘𝑤) = ( 𝐵𝑤))
29 ordelon 4305 . . . . . . . . . 10 ((Ord 𝑋𝐷𝑋) → 𝐷 ∈ On)
303, 8, 29syl2anc 408 . . . . . . . . 9 (𝜑𝐷 ∈ On)
31 eloni 4297 . . . . . . . . 9 (𝐷 ∈ On → Ord 𝐷)
3230, 31syl 14 . . . . . . . 8 (𝜑 → Ord 𝐷)
33 ordelss 4301 . . . . . . . 8 ((Ord 𝐷𝑤𝐷) → 𝑤𝐷)
3432, 33sylan 281 . . . . . . 7 ((𝜑𝑤𝐷) → 𝑤𝐷)
3512adantr 274 . . . . . . 7 ((𝜑𝑤𝐷) → dom 𝐵 = 𝐷)
3634, 35sseqtrrd 3136 . . . . . 6 ((𝜑𝑤𝐷) → 𝑤 ⊆ dom 𝐵)
37 fun2ssres 5166 . . . . . 6 ((Fun recs(𝐺) ∧ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ⊆ dom 𝐵) → (recs(𝐺) ↾ 𝑤) = ( 𝐵𝑤))
3824, 16, 36, 37mp3an2ani 1322 . . . . 5 ((𝜑𝑤𝐷) → (recs(𝐺) ↾ 𝑤) = ( 𝐵𝑤))
3938fveq2d 5425 . . . 4 ((𝜑𝑤𝐷) → (𝐺‘(recs(𝐺) ↾ 𝑤)) = (𝐺‘( 𝐵𝑤)))
4023, 28, 393eqtr3d 2180 . . 3 ((𝜑𝑤𝐷) → ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
4140ralrimiva 2505 . 2 (𝜑 → ∀𝑤𝐷 ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
42 fveq2 5421 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
43 reseq2 4814 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
4443fveq2d 5425 . . . 4 (𝑢 = 𝑤 → (𝐺‘( 𝐵𝑢)) = (𝐺‘( 𝐵𝑤)))
4542, 44eqeq12d 2154 . . 3 (𝑢 = 𝑤 → (( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)) ↔ ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤))))
4645cbvralv 2654 . 2 (∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)) ↔ ∀𝑤𝐷 ( 𝐵𝑤) = (𝐺‘( 𝐵𝑤)))
4741, 46sylibr 133 1 (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wex 1468  wcel 1480  {cab 2125  wral 2416  wrex 2417  cun 3069  wss 3071  {csn 3527  cop 3530   cuni 3736  Ord word 4284  Oncon0 4285  suc csuc 4287  dom cdm 4539  cres 4541  Fun wfun 5117   Fn wfn 5118  wf 5119  cfv 5123  recscrecs 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202
This theorem is referenced by:  tfrcllemex  6257
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