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Theorem tfrlemibfn 6047
Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6051. (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 6045 . . . . 5 (𝜑𝐵𝐴)
76unissd 3660 . . . 4 (𝜑 𝐵 𝐴)
81recsfval 6034 . . . 4 recs(𝐹) = 𝐴
97, 8syl6sseqr 3062 . . 3 (𝜑 𝐵 ⊆ recs(𝐹))
101tfrlem7 6036 . . 3 Fun recs(𝐹)
11 funss 4999 . . 3 ( 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun 𝐵))
129, 10, 11mpisyl 1378 . 2 (𝜑 → Fun 𝐵)
13 simpr3 949 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
142ad2antrr 472 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
154ad2antrr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑥 ∈ On)
16 simplr 497 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧𝑥)
17 onelon 4185 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
1815, 16, 17syl2anc 403 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧 ∈ On)
19 simpr1 947 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔 Fn 𝑧)
20 simpr2 948 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔𝐴)
211, 14, 18, 19, 20tfrlemisucfn 6043 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
22 dffn2 5128 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V)
2321, 22sylib 120 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V)
24 fssxp 5142 . . . . . . . . . . . . . . 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 4176 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → Ord 𝑥)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → Ord 𝑥)
28 ordsucss 4294 . . . . . . . . . . . . . . . 16 (Ord 𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
2927, 16, 28sylc 61 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → suc 𝑧𝑥)
30 xpss1 4516 . . . . . . . . . . . . . . 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 3024 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V))
33 vex 2618 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
34 vex 2618 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
352tfrlem3-2d 6031 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
3635simprd 112 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝑔) ∈ V)
37 opexg 4029 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
3834, 36, 37sylancr 405 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
39 snexg 3993 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
41 unexg 4242 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
4233, 40, 41sylancr 405 . . . . . . . . . . . . . . 15 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
43 elpwg 3423 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V)))
4632, 45mpbird 165 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V))
4713, 46eqeltrd 2161 . . . . . . . . . . 11 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∈ 𝒫 (𝑥 × V))
4847ex 113 . . . . . . . . . 10 ((𝜑𝑧𝑥) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
4948exlimdv 1744 . . . . . . . . 9 ((𝜑𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5049rexlimdva 2485 . . . . . . . 8 (𝜑 → (∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5150abssdv 3084 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
523, 51syl5eqss 3059 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝑥 × V))
53 sspwuni 3795 . . . . . 6 (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ 𝐵 ⊆ (𝑥 × V))
5452, 53sylib 120 . . . . 5 (𝜑 𝐵 ⊆ (𝑥 × V))
55 dmss 4603 . . . . 5 ( 𝐵 ⊆ (𝑥 × V) → dom 𝐵 ⊆ dom (𝑥 × V))
5654, 55syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝑥 × V))
57 dmxpss 4827 . . . 4 dom (𝑥 × V) ⊆ 𝑥
5856, 57syl6ss 3026 . . 3 (𝜑 → dom 𝐵𝑥)
591, 2, 3, 4, 5tfrlemibxssdm 6046 . . 3 (𝜑𝑥 ⊆ dom 𝐵)
6058, 59eqssd 3031 . 2 (𝜑 → dom 𝐵 = 𝑥)
61 df-fn 4984 . 2 ( 𝐵 Fn 𝑥 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝑥))
6212, 60, 61sylanbrc 408 1 (𝜑 𝐵 Fn 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922  wal 1285   = wceq 1287  wex 1424  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2615  cun 2986  wss 2988  𝒫 cpw 3415  {csn 3431  cop 3434   cuni 3636  Ord word 4163  Oncon0 4164  suc csuc 4166   × cxp 4409  dom cdm 4411  cres 4413  Fun wfun 4975   Fn wfn 4976  wf 4977  cfv 4981  recscrecs 6023
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-recs 6024
This theorem is referenced by:  tfrlemibex  6048  tfrlemiubacc  6049  tfrlemiex  6050
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