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Theorem tfrlemibfn 6191
Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6195. (Contributed by Jim Kingdon, 18-Mar-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
tfrlemibfn (𝜑 𝐵 Fn 𝑥)
Distinct variable groups:   𝑓,𝑔,,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑤,𝐵,𝑓,𝑔,,𝑧   𝜑,𝑔,,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemibfn
StepHypRef Expression
1 tfrlemisucfn.1 . . . . . 6 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 tfrlemisucfn.2 . . . . . 6 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
3 tfrlemi1.3 . . . . . 6 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
4 tfrlemi1.4 . . . . . 6 (𝜑𝑥 ∈ On)
5 tfrlemi1.5 . . . . . 6 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
61, 2, 3, 4, 5tfrlemibacc 6189 . . . . 5 (𝜑𝐵𝐴)
76unissd 3728 . . . 4 (𝜑 𝐵 𝐴)
81recsfval 6178 . . . 4 recs(𝐹) = 𝐴
97, 8sseqtrrdi 3114 . . 3 (𝜑 𝐵 ⊆ recs(𝐹))
101tfrlem7 6180 . . 3 Fun recs(𝐹)
11 funss 5110 . . 3 ( 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun 𝐵))
129, 10, 11mpisyl 1405 . 2 (𝜑 → Fun 𝐵)
13 simpr3 972 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
142ad2antrr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
154ad2antrr 477 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑥 ∈ On)
16 simplr 502 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧𝑥)
17 onelon 4274 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
1815, 16, 17syl2anc 406 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧 ∈ On)
19 simpr1 970 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔 Fn 𝑧)
20 simpr2 971 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔𝐴)
211, 14, 18, 19, 20tfrlemisucfn 6187 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
22 dffn2 5242 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V)
2321, 22sylib 121 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V)
24 fssxp 5258 . . . . . . . . . . . . . . 15 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (suc 𝑧 × V))
2523, 24syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (suc 𝑧 × V))
26 eloni 4265 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → Ord 𝑥)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → Ord 𝑥)
28 ordsucss 4388 . . . . . . . . . . . . . . . 16 (Ord 𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
2927, 16, 28sylc 62 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → suc 𝑧𝑥)
30 xpss1 4617 . . . . . . . . . . . . . . 15 (suc 𝑧𝑥 → (suc 𝑧 × V) ⊆ (𝑥 × V))
3129, 30syl 14 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (suc 𝑧 × V) ⊆ (𝑥 × V))
3225, 31sstrd 3075 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V))
33 vex 2661 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
34 vex 2661 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
352tfrlem3-2d 6175 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
3635simprd 113 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝑔) ∈ V)
37 opexg 4118 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
3834, 36, 37sylancr 408 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
39 snexg 4076 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
41 unexg 4332 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
4233, 40, 41sylancr 408 . . . . . . . . . . . . . . 15 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
43 elpwg 3486 . . . . . . . . . . . . . . 15 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V)))
4442, 43syl 14 . . . . . . . . . . . . . 14 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V)))
4544ad2antrr 477 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V)))
4632, 45mpbird 166 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V))
4713, 46eqeltrd 2192 . . . . . . . . . . 11 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∈ 𝒫 (𝑥 × V))
4847ex 114 . . . . . . . . . 10 ((𝜑𝑧𝑥) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
4948exlimdv 1773 . . . . . . . . 9 ((𝜑𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5049rexlimdva 2524 . . . . . . . 8 (𝜑 → (∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5150abssdv 3139 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
523, 51eqsstrid 3111 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝑥 × V))
53 sspwuni 3865 . . . . . 6 (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ 𝐵 ⊆ (𝑥 × V))
5452, 53sylib 121 . . . . 5 (𝜑 𝐵 ⊆ (𝑥 × V))
55 dmss 4706 . . . . 5 ( 𝐵 ⊆ (𝑥 × V) → dom 𝐵 ⊆ dom (𝑥 × V))
5654, 55syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝑥 × V))
57 dmxpss 4937 . . . 4 dom (𝑥 × V) ⊆ 𝑥
5856, 57sstrdi 3077 . . 3 (𝜑 → dom 𝐵𝑥)
591, 2, 3, 4, 5tfrlemibxssdm 6190 . . 3 (𝜑𝑥 ⊆ dom 𝐵)
6058, 59eqssd 3082 . 2 (𝜑 → dom 𝐵 = 𝑥)
61 df-fn 5094 . 2 ( 𝐵 Fn 𝑥 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝑥))
6212, 60, 61sylanbrc 411 1 (𝜑 𝐵 Fn 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945  wal 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2391  wrex 2392  Vcvv 2658  cun 3037  wss 3039  𝒫 cpw 3478  {csn 3495  cop 3498   cuni 3704  Ord word 4252  Oncon0 4253  suc csuc 4255   × cxp 4505  dom cdm 4507  cres 4509  Fun wfun 5085   Fn wfn 5086  wf 5087  cfv 5091  recscrecs 6167
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 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 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
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 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfrlemibex  6192  tfrlemiubacc  6193  tfrlemiex  6194
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