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Theorem dfac5lem5 9243
Description: Lemma for dfac5 9244. (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 9242 . 2 (𝜑 → ∃𝑦𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
5 simpr 473 . . . . . . . . . 10 ((𝑤 ≠ ∅ ∧ 𝑤) → 𝑤)
65a1i 11 . . . . . . . . 9 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → 𝑤))
7 ineq1 4017 . . . . . . . . . . . . 13 (𝑧 = ({𝑤} × 𝑤) → (𝑧𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
87eleq2d 2882 . . . . . . . . . . . 12 (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
98eubidv 2647 . . . . . . . . . . 11 (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
109rspccv 3510 . . . . . . . . . 10 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
111dfac5lem3 9241 . . . . . . . . . 10 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
12 dfac5lem1 9239 . . . . . . . . . 10 (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
1310, 11, 123imtr3g 286 . . . . . . . . 9 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
146, 13jcad 504 . . . . . . . 8 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → (𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
152eleq2i 2888 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ ( 𝐴𝑦))
16 elin 4006 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ ( 𝐴𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
171dfac5lem2 9240 . . . . . . . . . . . . 13 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))
1817anbi1i 612 . . . . . . . . . . . 12 ((⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ ((𝑤𝑔𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
19 anass 456 . . . . . . . . . . . 12 (((𝑤𝑔𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2018, 19bitri 266 . . . . . . . . . . 11 ((⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2115, 16, 203bitri 288 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2221eubii 2666 . . . . . . . . 9 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23 euanv 2709 . . . . . . . . 9 (∃!𝑔(𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ (𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2422, 23bitr2i 267 . . . . . . . 8 ((𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵)
2514, 24syl6ib 242 . . . . . . 7 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵))
26 euex 2667 . . . . . . . 8 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔𝑤, 𝑔⟩ ∈ 𝐵)
27 nfeu1 2653 . . . . . . . . . 10 𝑔∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵
28 nfv 2005 . . . . . . . . . 10 𝑔(𝐵𝑤) ∈ 𝑤
2927, 28nfim 1987 . . . . . . . . 9 𝑔(∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)
3021simprbi 486 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
3130simpld 484 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝐵𝑔𝑤)
32 tz6.12 6441 . . . . . . . . . . . . 13 ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵) → (𝐵𝑤) = 𝑔)
3332eleq1d 2881 . . . . . . . . . . . 12 ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵) → ((𝐵𝑤) ∈ 𝑤𝑔𝑤))
3433biimparc 467 . . . . . . . . . . 11 ((𝑔𝑤 ∧ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵)) → (𝐵𝑤) ∈ 𝑤)
3534exp32 409 . . . . . . . . . 10 (𝑔𝑤 → (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)))
3631, 35mpcom 38 . . . . . . . . 9 (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤))
3729, 36exlimi 2254 . . . . . . . 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 404 . . . . 5 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → (𝑤 → (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4140ralrimiv 3164 . . . 4 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤))
42 vex 3405 . . . . . . 7 𝑦 ∈ V
4342inex2 5008 . . . . . 6 ( 𝐴𝑦) ∈ V
442, 43eqeltri 2892 . . . . 5 𝐵 ∈ V
45 fveq1 6417 . . . . . . . 8 (𝑓 = 𝐵 → (𝑓𝑤) = (𝐵𝑤))
4645eleq1d 2881 . . . . . . 7 (𝑓 = 𝐵 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝐵𝑤) ∈ 𝑤))
4746imbi2d 331 . . . . . 6 (𝑓 = 𝐵 → ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4847ralbidv 3185 . . . . 5 (𝑓 = 𝐵 → (∀𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ↔ ∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4944, 48spcev 3504 . . . 4 (∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
5041, 49syl 17 . . 3 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
5150exlimiv 2021 . 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 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2157  ∃!weu 2641  {cab 2803  wne 2989  wral 3107  wrex 3108  Vcvv 3402  cin 3779  c0 4127  {csn 4381  cop 4387   cuni 4641   × cxp 5322  cfv 6111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-xp 5330  df-rel 5331  df-cnv 5332  df-dm 5334  df-rn 5335  df-iota 6074  df-fv 6119
This theorem is referenced by:  dfac5  9244
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