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

Theorem tfrlemiubacc 6077
Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6079). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
tfrlemi1.3 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
tfrlemi1.4 (𝜑𝑥 ∈ On)
tfrlemi1.5 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
Assertion
Ref Expression
tfrlemiubacc (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
Distinct variable groups:   𝑓,𝑔,,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,,𝑢,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑢,𝐵,𝑤,𝑓,𝑔,,𝑧   𝜑,𝑔,,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑢,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemiubacc
StepHypRef Expression
1 tfrlemisucfn.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 tfrlemisucfn.2 . . . . . . . . 9 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
3 tfrlemi1.3 . . . . . . . . 9 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
4 tfrlemi1.4 . . . . . . . . 9 (𝜑𝑥 ∈ On)
5 tfrlemi1.5 . . . . . . . . 9 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
61, 2, 3, 4, 5tfrlemibfn 6075 . . . . . . . 8 (𝜑 𝐵 Fn 𝑥)
7 fndm 5099 . . . . . . . 8 ( 𝐵 Fn 𝑥 → dom 𝐵 = 𝑥)
86, 7syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝑥)
91, 2, 3, 4, 5tfrlemibacc 6073 . . . . . . . . . 10 (𝜑𝐵𝐴)
109unissd 3672 . . . . . . . . 9 (𝜑 𝐵 𝐴)
111recsfval 6062 . . . . . . . . 9 recs(𝐹) = 𝐴
1210, 11syl6sseqr 3071 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐹))
13 dmss 4623 . . . . . . . 8 ( 𝐵 ⊆ recs(𝐹) → dom 𝐵 ⊆ dom recs(𝐹))
1412, 13syl 14 . . . . . . 7 (𝜑 → dom 𝐵 ⊆ dom recs(𝐹))
158, 14eqsstr3d 3059 . . . . . 6 (𝜑𝑥 ⊆ dom recs(𝐹))
1615sselda 3023 . . . . 5 ((𝜑𝑤𝑥) → 𝑤 ∈ dom recs(𝐹))
171tfrlem9 6066 . . . . 5 (𝑤 ∈ dom recs(𝐹) → (recs(𝐹)‘𝑤) = (𝐹‘(recs(𝐹) ↾ 𝑤)))
1816, 17syl 14 . . . 4 ((𝜑𝑤𝑥) → (recs(𝐹)‘𝑤) = (𝐹‘(recs(𝐹) ↾ 𝑤)))
191tfrlem7 6064 . . . . . 6 Fun recs(𝐹)
2019a1i 9 . . . . 5 ((𝜑𝑤𝑥) → Fun recs(𝐹))
2112adantr 270 . . . . 5 ((𝜑𝑤𝑥) → 𝐵 ⊆ recs(𝐹))
228eleq2d 2157 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝑥))
2322biimpar 291 . . . . 5 ((𝜑𝑤𝑥) → 𝑤 ∈ dom 𝐵)
24 funssfv 5314 . . . . 5 ((Fun recs(𝐹) ∧ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom 𝐵) → (recs(𝐹)‘𝑤) = ( 𝐵𝑤))
2520, 21, 23, 24syl3anc 1174 . . . 4 ((𝜑𝑤𝑥) → (recs(𝐹)‘𝑤) = ( 𝐵𝑤))
26 eloni 4193 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
274, 26syl 14 . . . . . . . 8 (𝜑 → Ord 𝑥)
28 ordelss 4197 . . . . . . . 8 ((Ord 𝑥𝑤𝑥) → 𝑤𝑥)
2927, 28sylan 277 . . . . . . 7 ((𝜑𝑤𝑥) → 𝑤𝑥)
308adantr 270 . . . . . . 7 ((𝜑𝑤𝑥) → dom 𝐵 = 𝑥)
3129, 30sseqtr4d 3061 . . . . . 6 ((𝜑𝑤𝑥) → 𝑤 ⊆ dom 𝐵)
32 fun2ssres 5043 . . . . . 6 ((Fun recs(𝐹) ∧ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom 𝐵) → (recs(𝐹) ↾ 𝑤) = ( 𝐵𝑤))
3320, 21, 31, 32syl3anc 1174 . . . . 5 ((𝜑𝑤𝑥) → (recs(𝐹) ↾ 𝑤) = ( 𝐵𝑤))
3433fveq2d 5293 . . . 4 ((𝜑𝑤𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘( 𝐵𝑤)))
3518, 25, 343eqtr3d 2128 . . 3 ((𝜑𝑤𝑥) → ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
3635ralrimiva 2446 . 2 (𝜑 → ∀𝑤𝑥 ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
37 fveq2 5289 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
38 reseq2 4696 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
3938fveq2d 5293 . . . 4 (𝑢 = 𝑤 → (𝐹‘( 𝐵𝑢)) = (𝐹‘( 𝐵𝑤)))
4037, 39eqeq12d 2102 . . 3 (𝑢 = 𝑤 → (( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)) ↔ ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤))))
4140cbvralv 2590 . 2 (∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)) ↔ ∀𝑤𝑥 ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
4236, 41sylibr 132 1 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924  wal 1287   = wceq 1289  wex 1426  wcel 1438  {cab 2074  wral 2359  wrex 2360  Vcvv 2619  cun 2995  wss 2997  {csn 3441  cop 3444   cuni 3648  Ord word 4180  Oncon0 4181  dom cdm 4428  cres 4430  Fun wfun 4996   Fn wfn 4997  cfv 5002  recscrecs 6051
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-recs 6052
This theorem is referenced by:  tfrlemiex  6078
  Copyright terms: Public domain W3C validator