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Theorem tfrlemibfn 6307
Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6311. (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 6305 . . . . 5 (𝜑𝐵𝐴)
76unissd 3820 . . . 4 (𝜑 𝐵 𝐴)
81recsfval 6294 . . . 4 recs(𝐹) = 𝐴
97, 8sseqtrrdi 3196 . . 3 (𝜑 𝐵 ⊆ recs(𝐹))
101tfrlem7 6296 . . 3 Fun recs(𝐹)
11 funss 5217 . . 3 ( 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun 𝐵))
129, 10, 11mpisyl 1439 . 2 (𝜑 → Fun 𝐵)
13 simpr3 1000 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
142ad2antrr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
154ad2antrr 485 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑥 ∈ On)
16 simplr 525 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧𝑥)
17 onelon 4369 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
1815, 16, 17syl2anc 409 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧 ∈ On)
19 simpr1 998 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔 Fn 𝑧)
20 simpr2 999 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔𝐴)
211, 14, 18, 19, 20tfrlemisucfn 6303 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
22 dffn2 5349 . . . . . . . . . . . . . . . 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 5365 . . . . . . . . . . . . . . 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 4360 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → Ord 𝑥)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → Ord 𝑥)
28 ordsucss 4488 . . . . . . . . . . . . . . . 16 (Ord 𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
2927, 16, 28sylc 62 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → suc 𝑧𝑥)
30 xpss1 4721 . . . . . . . . . . . . . . 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 3157 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V))
33 vex 2733 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
34 vex 2733 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
352tfrlem3-2d 6291 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
3635simprd 113 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝑔) ∈ V)
37 opexg 4213 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
3834, 36, 37sylancr 412 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
39 snexg 4170 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
41 unexg 4428 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
4233, 40, 41sylancr 412 . . . . . . . . . . . . . . 15 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
43 elpwg 3574 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 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 2247 . . . . . . . . . . 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 1812 . . . . . . . . 9 ((𝜑𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5049rexlimdva 2587 . . . . . . . 8 (𝜑 → (∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5150abssdv 3221 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
523, 51eqsstrid 3193 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝑥 × V))
53 sspwuni 3957 . . . . . 6 (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ 𝐵 ⊆ (𝑥 × V))
5452, 53sylib 121 . . . . 5 (𝜑 𝐵 ⊆ (𝑥 × V))
55 dmss 4810 . . . . 5 ( 𝐵 ⊆ (𝑥 × V) → dom 𝐵 ⊆ dom (𝑥 × V))
5654, 55syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝑥 × V))
57 dmxpss 5041 . . . 4 dom (𝑥 × V) ⊆ 𝑥
5856, 57sstrdi 3159 . . 3 (𝜑 → dom 𝐵𝑥)
591, 2, 3, 4, 5tfrlemibxssdm 6306 . . 3 (𝜑𝑥 ⊆ dom 𝐵)
6058, 59eqssd 3164 . 2 (𝜑 → dom 𝐵 = 𝑥)
61 df-fn 5201 . 2 ( 𝐵 Fn 𝑥 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝑥))
6212, 60, 61sylanbrc 415 1 (𝜑 𝐵 Fn 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973  wal 1346   = wceq 1348  wex 1485  wcel 2141  {cab 2156  wral 2448  wrex 2449  Vcvv 2730  cun 3119  wss 3121  𝒫 cpw 3566  {csn 3583  cop 3586   cuni 3796  Ord word 4347  Oncon0 4348  suc csuc 4350   × cxp 4609  dom cdm 4611  cres 4613  Fun wfun 5192   Fn wfn 5193  wf 5194  cfv 5198  recscrecs 6283
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-recs 6284
This theorem is referenced by:  tfrlemibex  6308  tfrlemiubacc  6309  tfrlemiex  6310
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