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Theorem tfrlemibfn 5972
 Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 5976. (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 5970 . . . . 5 (𝜑𝐵𝐴)
76unissd 3631 . . . 4 (𝜑 𝐵 𝐴)
81recsfval 5961 . . . 4 recs(𝐹) = 𝐴
97, 8syl6sseqr 3019 . . 3 (𝜑 𝐵 ⊆ recs(𝐹))
101tfrlem7 5963 . . 3 Fun recs(𝐹)
11 funss 4947 . . 3 ( 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun 𝐵))
129, 10, 11mpisyl 1351 . 2 (𝜑 → Fun 𝐵)
13 simpr3 923 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
142ad2antrr 465 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
154ad2antrr 465 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑥 ∈ On)
16 simplr 490 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧𝑥)
17 onelon 4148 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
1815, 16, 17syl2anc 397 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧 ∈ On)
19 simpr1 921 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔 Fn 𝑧)
20 simpr2 922 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔𝐴)
211, 14, 18, 19, 20tfrlemisucfn 5968 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
22 dffn2 5074 . . . . . . . . . . . . . . . 16 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V)
2321, 22sylib 131 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}):suc 𝑧⟶V)
24 fssxp 5085 . . . . . . . . . . . . . . 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 4139 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → Ord 𝑥)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → Ord 𝑥)
28 ordsucss 4257 . . . . . . . . . . . . . . . 16 (Ord 𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
2927, 16, 28sylc 60 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → suc 𝑧𝑥)
30 xpss1 4475 . . . . . . . . . . . . . . 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 2982 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V))
33 vex 2577 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
34 vex 2577 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
352tfrlem3-2d 5958 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
3635simprd 111 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝑔) ∈ V)
37 opexg 3991 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
3834, 36, 37sylancr 399 . . . . . . . . . . . . . . . . 17 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
39 snexg 3963 . . . . . . . . . . . . . . . . 17 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
41 unexg 4205 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
4233, 40, 41sylancr 399 . . . . . . . . . . . . . . 15 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
43 elpwg 3394 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ⊆ (𝑥 × V)))
4632, 45mpbird 160 . . . . . . . . . . . 12 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝒫 (𝑥 × V))
4713, 46eqeltrd 2130 . . . . . . . . . . 11 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∈ 𝒫 (𝑥 × V))
4847ex 112 . . . . . . . . . 10 ((𝜑𝑧𝑥) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
4948exlimdv 1716 . . . . . . . . 9 ((𝜑𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5049rexlimdva 2450 . . . . . . . 8 (𝜑 → (∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → ∈ 𝒫 (𝑥 × V)))
5150abssdv 3041 . . . . . . 7 (𝜑 → { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
523, 51syl5eqss 3016 . . . . . 6 (𝜑𝐵 ⊆ 𝒫 (𝑥 × V))
53 sspwuni 3766 . . . . . 6 (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ 𝐵 ⊆ (𝑥 × V))
5452, 53sylib 131 . . . . 5 (𝜑 𝐵 ⊆ (𝑥 × V))
55 dmss 4561 . . . . 5 ( 𝐵 ⊆ (𝑥 × V) → dom 𝐵 ⊆ dom (𝑥 × V))
5654, 55syl 14 . . . 4 (𝜑 → dom 𝐵 ⊆ dom (𝑥 × V))
57 dmxpss 4780 . . . 4 dom (𝑥 × V) ⊆ 𝑥
5856, 57syl6ss 2984 . . 3 (𝜑 → dom 𝐵𝑥)
591, 2, 3, 4, 5tfrlemibxssdm 5971 . . 3 (𝜑𝑥 ⊆ dom 𝐵)
6058, 59eqssd 2989 . 2 (𝜑 → dom 𝐵 = 𝑥)
61 df-fn 4932 . 2 ( 𝐵 Fn 𝑥 ↔ (Fun 𝐵 ∧ dom 𝐵 = 𝑥))
6212, 60, 61sylanbrc 402 1 (𝜑 𝐵 Fn 𝑥)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∧ w3a 896  ∀wal 1257   = wceq 1259  ∃wex 1397   ∈ wcel 1409  {cab 2042  ∀wral 2323  ∃wrex 2324  Vcvv 2574   ∪ cun 2942   ⊆ wss 2944  𝒫 cpw 3386  {csn 3402  ⟨cop 3405  ∪ cuni 3607  Ord word 4126  Oncon0 4127  suc csuc 4129   × cxp 4370  dom cdm 4372   ↾ cres 4374  Fun wfun 4923   Fn wfn 4924  ⟶wf 4925  ‘cfv 4929  recscrecs 5949 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937  df-recs 5950 This theorem is referenced by:  tfrlemibex  5973  tfrlemiubacc  5974  tfrlemiex  5975
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