ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrcllemex GIF version

Theorem tfrcllemex 6354
Description: Lemma for tfrcl 6358. (Contributed by Jim Kingdon, 26-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
tfrcllemex (𝜑 → ∃𝑓(𝑓:𝐷𝑆 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))))
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,𝑓,𝑔,,𝑧   𝑢,𝐵,𝑓   𝑤,𝐵,𝑔,𝑧   𝐷,,𝑧   𝑢,𝐷,𝑤   𝑦,𝑤   ,𝐺,𝑧   𝑢,𝐺,𝑤   𝑆,𝑔,,𝑧   𝑧,𝑋   𝜑,𝑤
Allowed substitution hints:   𝜑(𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦)   𝑆(𝑤,𝑢)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓,𝑔,)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑢,𝑔,)

Proof of Theorem tfrcllemex
StepHypRef Expression
1 tfrcl.f . . . 4 𝐹 = recs(𝐺)
2 tfrcl.g . . . 4 (𝜑 → Fun 𝐺)
3 tfrcl.x . . . 4 (𝜑 → Ord 𝑋)
4 tfrcl.ex . . . 4 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
5 tfrcllemsucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
6 tfrcllembacc.3 . . . 4 𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}
7 tfrcllembacc.u . . . 4 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfrcllembacc.4 . . . 4 (𝜑𝐷𝑋)
9 tfrcllembacc.5 . . . 4 (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))
101, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembex 6352 . . 3 (𝜑𝐵 ∈ V)
11 uniexg 4435 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
1210, 11syl 14 . 2 (𝜑 𝐵 ∈ V)
131, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembfn 6351 . . 3 (𝜑 𝐵:𝐷𝑆)
141, 2, 3, 4, 5, 6, 7, 8, 9tfrcllemubacc 6353 . . 3 (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
1513, 14jca 306 . 2 (𝜑 → ( 𝐵:𝐷𝑆 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
16 feq1 5343 . . . 4 (𝑓 = 𝐵 → (𝑓:𝐷𝑆 𝐵:𝐷𝑆))
17 fveq1 5509 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
18 reseq1 4896 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1918fveq2d 5514 . . . . . 6 (𝑓 = 𝐵 → (𝐺‘(𝑓𝑢)) = (𝐺‘( 𝐵𝑢)))
2017, 19eqeq12d 2192 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐺‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
2120ralbidv 2477 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢)) ↔ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))))
2216, 21anbi12d 473 . . 3 (𝑓 = 𝐵 → ((𝑓:𝐷𝑆 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))) ↔ ( 𝐵:𝐷𝑆 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))))
2322spcegv 2825 . 2 ( 𝐵 ∈ V → (( 𝐵:𝐷𝑆 ∧ ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢))) → ∃𝑓(𝑓:𝐷𝑆 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢)))))
2412, 15, 23sylc 62 1 (𝜑 → ∃𝑓(𝑓:𝐷𝑆 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2737  cun 3127  {csn 3591  cop 3594   cuni 3807  Ord word 4358  suc csuc 4361  cres 4624  Fun wfun 5205  wf 5207  cfv 5211  recscrecs 6298
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-recs 6299
This theorem is referenced by:  tfrcllemaccex  6355
  Copyright terms: Public domain W3C validator