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Theorem tfrlemibfn 6193
Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6197. (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 6191 . . . . 5 (𝜑𝐵𝐴)
76unissd 3730 . . . 4 (𝜑 𝐵 𝐴)
81recsfval 6180 . . . 4 recs(𝐹) = 𝐴
97, 8sseqtrrdi 3116 . . 3 (𝜑 𝐵 ⊆ recs(𝐹))
101tfrlem7 6182 . . 3 Fun recs(𝐹)
11 funss 5112 . . 3 ( 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun 𝐵))
129, 10, 11mpisyl 1407 . 2 (𝜑 → Fun 𝐵)
13 simpr3 974 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
142ad2antrr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
154ad2antrr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑥 ∈ On)
16 simplr 504 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧𝑥)
17 onelon 4276 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
1815, 16, 17syl2anc 408 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧 ∈ On)
19 simpr1 972 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔 Fn 𝑧)
20 simpr2 973 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔𝐴)
211, 14, 18, 19, 20tfrlemisucfn 6189 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
22 dffn2 5244 . . . . . . . . . . . . . . . 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 5260 . . . . . . . . . . . . . . 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 4267 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → Ord 𝑥)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → Ord 𝑥)
28 ordsucss 4390 . . . . . . . . . . . . . . . 16 (Ord 𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
2927, 16, 28sylc 62 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → suc 𝑧𝑥)
30 xpss1 4619 . . . . . . . . . . . . . . 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 3077 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V))
33 vex 2663 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
34 vex 2663 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
352tfrlem3-2d 6177 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
3635simprd 113 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝑔) ∈ V)
37 opexg 4120 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
3834, 36, 37sylancr 410 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
39 snexg 4078 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
41 unexg 4334 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
4233, 40, 41sylancr 410 . . . . . . . . . . . . . . 15 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
43 elpwg 3488 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 2194 . . . . . . . . . . 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 1775 . . . . . . . . 9 ((𝜑𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5049rexlimdva 2526 . . . . . . . 8 (𝜑 → (∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5150abssdv 3141 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
523, 51eqsstrid 3113 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝑥 × V))
53 sspwuni 3867 . . . . . 6 (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ 𝐵 ⊆ (𝑥 × V))
5452, 53sylib 121 . . . . 5 (𝜑 𝐵 ⊆ (𝑥 × V))
55 dmss 4708 . . . . 5 ( 𝐵 ⊆ (𝑥 × V) → dom 𝐵 ⊆ dom (𝑥 × V))
5654, 55syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝑥 × V))
57 dmxpss 4939 . . . 4 dom (𝑥 × V) ⊆ 𝑥
5856, 57sstrdi 3079 . . 3 (𝜑 → dom 𝐵𝑥)
591, 2, 3, 4, 5tfrlemibxssdm 6192 . . 3 (𝜑𝑥 ⊆ dom 𝐵)
6058, 59eqssd 3084 . 2 (𝜑 → dom 𝐵 = 𝑥)
61 df-fn 5096 . 2 ( 𝐵 Fn 𝑥 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝑥))
6212, 60, 61sylanbrc 413 1 (𝜑 𝐵 Fn 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947  wal 1314   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  cun 3039  wss 3041  𝒫 cpw 3480  {csn 3497  cop 3500   cuni 3706  Ord word 4254  Oncon0 4255  suc csuc 4257   × cxp 4507  dom cdm 4509  cres 4511  Fun wfun 5087   Fn wfn 5088  wf 5089  cfv 5093  recscrecs 6169
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfrlemibex  6194  tfrlemiubacc  6195  tfrlemiex  6196
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