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Theorem tfrlemiubacc 6181
Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6183). (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 6179 . . . . . . . 8 (𝜑 𝐵 Fn 𝑥)
7 fndm 5180 . . . . . . . 8 ( 𝐵 Fn 𝑥 → dom 𝐵 = 𝑥)
86, 7syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝑥)
91, 2, 3, 4, 5tfrlemibacc 6177 . . . . . . . . . 10 (𝜑𝐵𝐴)
109unissd 3726 . . . . . . . . 9 (𝜑 𝐵 𝐴)
111recsfval 6166 . . . . . . . . 9 recs(𝐹) = 𝐴
1210, 11syl6sseqr 3112 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐹))
13 dmss 4698 . . . . . . . 8 ( 𝐵 ⊆ recs(𝐹) → dom 𝐵 ⊆ dom recs(𝐹))
1412, 13syl 14 . . . . . . 7 (𝜑 → dom 𝐵 ⊆ dom recs(𝐹))
158, 14eqsstrrd 3100 . . . . . 6 (𝜑𝑥 ⊆ dom recs(𝐹))
1615sselda 3063 . . . . 5 ((𝜑𝑤𝑥) → 𝑤 ∈ dom recs(𝐹))
171tfrlem9 6170 . . . . 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 6168 . . . . . 6 Fun recs(𝐹)
2019a1i 9 . . . . 5 ((𝜑𝑤𝑥) → Fun recs(𝐹))
2112adantr 272 . . . . 5 ((𝜑𝑤𝑥) → 𝐵 ⊆ recs(𝐹))
228eleq2d 2184 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝑥))
2322biimpar 293 . . . . 5 ((𝜑𝑤𝑥) → 𝑤 ∈ dom 𝐵)
24 funssfv 5401 . . . . 5 ((Fun recs(𝐹) ∧ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom 𝐵) → (recs(𝐹)‘𝑤) = ( 𝐵𝑤))
2520, 21, 23, 24syl3anc 1199 . . . 4 ((𝜑𝑤𝑥) → (recs(𝐹)‘𝑤) = ( 𝐵𝑤))
26 eloni 4257 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
274, 26syl 14 . . . . . . . 8 (𝜑 → Ord 𝑥)
28 ordelss 4261 . . . . . . . 8 ((Ord 𝑥𝑤𝑥) → 𝑤𝑥)
2927, 28sylan 279 . . . . . . 7 ((𝜑𝑤𝑥) → 𝑤𝑥)
308adantr 272 . . . . . . 7 ((𝜑𝑤𝑥) → dom 𝐵 = 𝑥)
3129, 30sseqtr4d 3102 . . . . . 6 ((𝜑𝑤𝑥) → 𝑤 ⊆ dom 𝐵)
32 fun2ssres 5124 . . . . . 6 ((Fun recs(𝐹) ∧ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom 𝐵) → (recs(𝐹) ↾ 𝑤) = ( 𝐵𝑤))
3320, 21, 31, 32syl3anc 1199 . . . . 5 ((𝜑𝑤𝑥) → (recs(𝐹) ↾ 𝑤) = ( 𝐵𝑤))
3433fveq2d 5379 . . . 4 ((𝜑𝑤𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘( 𝐵𝑤)))
3518, 25, 343eqtr3d 2155 . . 3 ((𝜑𝑤𝑥) → ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
3635ralrimiva 2479 . 2 (𝜑 → ∀𝑤𝑥 ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
37 fveq2 5375 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
38 reseq2 4772 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
3938fveq2d 5379 . . . 4 (𝑢 = 𝑤 → (𝐹‘( 𝐵𝑢)) = (𝐹‘( 𝐵𝑤)))
4037, 39eqeq12d 2129 . . 3 (𝑢 = 𝑤 → (( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)) ↔ ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤))))
4140cbvralv 2628 . 2 (∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)) ↔ ∀𝑤𝑥 ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
4236, 41sylibr 133 1 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945  wal 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2390  wrex 2391  Vcvv 2657  cun 3035  wss 3037  {csn 3493  cop 3496   cuni 3702  Ord word 4244  Oncon0 4245  dom cdm 4499  cres 4501  Fun wfun 5075   Fn wfn 5076  cfv 5081  recscrecs 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-recs 6156
This theorem is referenced by:  tfrlemiex  6182
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