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Theorem tfrlemiubacc 5944
Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 5946). (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 5942 . . . . . . . 8 (𝜑 𝐵 Fn 𝑥)
7 fndm 4998 . . . . . . . 8 ( 𝐵 Fn 𝑥 → dom 𝐵 = 𝑥)
86, 7syl 14 . . . . . . 7 (𝜑 → dom 𝐵 = 𝑥)
91, 2, 3, 4, 5tfrlemibacc 5940 . . . . . . . . . 10 (𝜑𝐵𝐴)
109unissd 3604 . . . . . . . . 9 (𝜑 𝐵 𝐴)
111recsfval 5931 . . . . . . . . 9 recs(𝐹) = 𝐴
1210, 11syl6sseqr 2992 . . . . . . . 8 (𝜑 𝐵 ⊆ recs(𝐹))
13 dmss 4534 . . . . . . . 8 ( 𝐵 ⊆ recs(𝐹) → dom 𝐵 ⊆ dom recs(𝐹))
1412, 13syl 14 . . . . . . 7 (𝜑 → dom 𝐵 ⊆ dom recs(𝐹))
158, 14eqsstr3d 2980 . . . . . 6 (𝜑𝑥 ⊆ dom recs(𝐹))
1615sselda 2945 . . . . 5 ((𝜑𝑤𝑥) → 𝑤 ∈ dom recs(𝐹))
171tfrlem9 5935 . . . . 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 5933 . . . . . 6 Fun recs(𝐹)
2019a1i 9 . . . . 5 ((𝜑𝑤𝑥) → Fun recs(𝐹))
2112adantr 261 . . . . 5 ((𝜑𝑤𝑥) → 𝐵 ⊆ recs(𝐹))
228eleq2d 2107 . . . . . 6 (𝜑 → (𝑤 ∈ dom 𝐵𝑤𝑥))
2322biimpar 281 . . . . 5 ((𝜑𝑤𝑥) → 𝑤 ∈ dom 𝐵)
24 funssfv 5199 . . . . 5 ((Fun recs(𝐹) ∧ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom 𝐵) → (recs(𝐹)‘𝑤) = ( 𝐵𝑤))
2520, 21, 23, 24syl3anc 1135 . . . 4 ((𝜑𝑤𝑥) → (recs(𝐹)‘𝑤) = ( 𝐵𝑤))
26 eloni 4112 . . . . . . . . 9 (𝑥 ∈ On → Ord 𝑥)
274, 26syl 14 . . . . . . . 8 (𝜑 → Ord 𝑥)
28 ordelss 4116 . . . . . . . 8 ((Ord 𝑥𝑤𝑥) → 𝑤𝑥)
2927, 28sylan 267 . . . . . . 7 ((𝜑𝑤𝑥) → 𝑤𝑥)
308adantr 261 . . . . . . 7 ((𝜑𝑤𝑥) → dom 𝐵 = 𝑥)
3129, 30sseqtr4d 2982 . . . . . 6 ((𝜑𝑤𝑥) → 𝑤 ⊆ dom 𝐵)
32 fun2ssres 4943 . . . . . 6 ((Fun recs(𝐹) ∧ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom 𝐵) → (recs(𝐹) ↾ 𝑤) = ( 𝐵𝑤))
3320, 21, 31, 32syl3anc 1135 . . . . 5 ((𝜑𝑤𝑥) → (recs(𝐹) ↾ 𝑤) = ( 𝐵𝑤))
3433fveq2d 5182 . . . 4 ((𝜑𝑤𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘( 𝐵𝑤)))
3518, 25, 343eqtr3d 2080 . . 3 ((𝜑𝑤𝑥) → ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
3635ralrimiva 2392 . 2 (𝜑 → ∀𝑤𝑥 ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
37 fveq2 5178 . . . 4 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
38 reseq2 4607 . . . . 5 (𝑢 = 𝑤 → ( 𝐵𝑢) = ( 𝐵𝑤))
3938fveq2d 5182 . . . 4 (𝑢 = 𝑤 → (𝐹‘( 𝐵𝑢)) = (𝐹‘( 𝐵𝑤)))
4037, 39eqeq12d 2054 . . 3 (𝑢 = 𝑤 → (( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)) ↔ ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤))))
4140cbvralv 2533 . 2 (∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)) ↔ ∀𝑤𝑥 ( 𝐵𝑤) = (𝐹‘( 𝐵𝑤)))
4236, 41sylibr 137 1 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885  wal 1241   = wceq 1243  wex 1381  wcel 1393  {cab 2026  wral 2306  wrex 2307  Vcvv 2557  cun 2915  wss 2917  {csn 3375  cop 3378   cuni 3580  Ord word 4099  Oncon0 4100  dom cdm 4345  cres 4347  Fun wfun 4896   Fn wfn 4897  cfv 4902  recscrecs 5919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-recs 5920
This theorem is referenced by:  tfrlemiex  5945
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