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Theorem dfac5lem5 10166
Description: Lemma for dfac5 10167. (Contributed by NM, 12-Apr-2004.)
Hypotheses
Ref Expression
dfac5lem.1 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
dfac5lem.2 𝐵 = ( 𝐴𝑦)
dfac5lem.3 (𝜑 ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
Assertion
Ref Expression
dfac5lem5 (𝜑 → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
Distinct variable groups:   𝑥,𝑓,𝑧,𝑦,𝑤,𝑣,𝑢,𝑡,   𝑧,𝐵,𝑤,𝑓   𝑥,𝐴,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,)   𝐴(𝑣,𝑢,𝑡,𝑓,)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑡,)

Proof of Theorem dfac5lem5
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 dfac5lem.1 . . 3 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
2 dfac5lem.2 . . 3 𝐵 = ( 𝐴𝑦)
3 dfac5lem.3 . . 3 (𝜑 ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
41, 2, 3dfac5lem4 10165 . 2 (𝜑 → ∃𝑦𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
5 simpr 483 . . . . . . . . . 10 ((𝑤 ≠ ∅ ∧ 𝑤) → 𝑤)
65a1i 11 . . . . . . . . 9 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → 𝑤))
7 ineq1 4205 . . . . . . . . . . . . 13 (𝑧 = ({𝑤} × 𝑤) → (𝑧𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
87eleq2d 2811 . . . . . . . . . . . 12 (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
98eubidv 2574 . . . . . . . . . . 11 (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
109rspccv 3604 . . . . . . . . . 10 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
111dfac5lem3 10164 . . . . . . . . . 10 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
12 dfac5lem1 10162 . . . . . . . . . 10 (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
1310, 11, 123imtr3g 294 . . . . . . . . 9 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
146, 13jcad 511 . . . . . . . 8 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → (𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
152eleq2i 2817 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ ( 𝐴𝑦))
16 elin 3962 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ ( 𝐴𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
171dfac5lem2 10163 . . . . . . . . . . . . 13 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))
1817anbi1i 622 . . . . . . . . . . . 12 ((⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ ((𝑤𝑔𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
19 anass 467 . . . . . . . . . . . 12 (((𝑤𝑔𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2018, 19bitri 274 . . . . . . . . . . 11 ((⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2115, 16, 203bitri 296 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2221eubii 2573 . . . . . . . . 9 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23 euanv 2612 . . . . . . . . 9 (∃!𝑔(𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ (𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2422, 23bitr2i 275 . . . . . . . 8 ((𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵)
2514, 24imbitrdi 250 . . . . . . 7 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵))
26 euex 2565 . . . . . . . 8 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔𝑤, 𝑔⟩ ∈ 𝐵)
27 nfeu1 2576 . . . . . . . . . 10 𝑔∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵
28 nfv 1909 . . . . . . . . . 10 𝑔(𝐵𝑤) ∈ 𝑤
2927, 28nfim 1891 . . . . . . . . 9 𝑔(∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)
3021simprbi 495 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
3130simpld 493 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝐵𝑔𝑤)
32 tz6.12 6925 . . . . . . . . . . . . 13 ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵) → (𝐵𝑤) = 𝑔)
3332eleq1d 2810 . . . . . . . . . . . 12 ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵) → ((𝐵𝑤) ∈ 𝑤𝑔𝑤))
3433biimparc 478 . . . . . . . . . . 11 ((𝑔𝑤 ∧ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵)) → (𝐵𝑤) ∈ 𝑤)
3534exp32 419 . . . . . . . . . 10 (𝑔𝑤 → (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)))
3631, 35mpcom 38 . . . . . . . . 9 (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤))
3729, 36exlimi 2205 . . . . . . . 8 (∃𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤))
3826, 37mpcom 38 . . . . . . 7 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)
3925, 38syl6 35 . . . . . 6 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → (𝐵𝑤) ∈ 𝑤))
4039expcomd 415 . . . . 5 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → (𝑤 → (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4140ralrimiv 3134 . . . 4 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤))
42 vex 3465 . . . . . . 7 𝑦 ∈ V
4342inex2 5322 . . . . . 6 ( 𝐴𝑦) ∈ V
442, 43eqeltri 2821 . . . . 5 𝐵 ∈ V
45 fveq1 6899 . . . . . . . 8 (𝑓 = 𝐵 → (𝑓𝑤) = (𝐵𝑤))
4645eleq1d 2810 . . . . . . 7 (𝑓 = 𝐵 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝐵𝑤) ∈ 𝑤))
4746imbi2d 339 . . . . . 6 (𝑓 = 𝐵 → ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4847ralbidv 3167 . . . . 5 (𝑓 = 𝐵 → (∀𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ↔ ∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4944, 48spcev 3591 . . . 4 (∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
5041, 49syl 17 . . 3 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
5150exlimiv 1925 . 2 (∃𝑦𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
524, 51syl 17 1 (𝜑 → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2556  {cab 2702  wne 2929  wral 3050  wrex 3059  Vcvv 3461  cin 3945  c0 4324  {csn 4632  cop 4638   cuni 4912   × cxp 5679  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5687  df-rel 5688  df-cnv 5689  df-dm 5691  df-rn 5692  df-iota 6505  df-fv 6561
This theorem is referenced by:  dfac5  10167
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