MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac5lem5 Structured version   Visualization version   GIF version

Theorem dfac5lem5 9160
Description: Lemma for dfac5 9161. (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 9159 . 2 (𝜑 → ∃𝑦𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
5 simpr 479 . . . . . . . . . 10 ((𝑤 ≠ ∅ ∧ 𝑤) → 𝑤)
65a1i 11 . . . . . . . . 9 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → 𝑤))
7 ineq1 3950 . . . . . . . . . . . . 13 (𝑧 = ({𝑤} × 𝑤) → (𝑧𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
87eleq2d 2825 . . . . . . . . . . . 12 (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
98eubidv 2627 . . . . . . . . . . 11 (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
109rspccv 3446 . . . . . . . . . 10 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
111dfac5lem3 9158 . . . . . . . . . 10 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
12 dfac5lem1 9156 . . . . . . . . . 10 (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
1310, 11, 123imtr3g 284 . . . . . . . . 9 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
146, 13jcad 556 . . . . . . . 8 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → (𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
152eleq2i 2831 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ ( 𝐴𝑦))
16 elin 3939 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ ( 𝐴𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
171dfac5lem2 9157 . . . . . . . . . . . . 13 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))
1817anbi1i 733 . . . . . . . . . . . 12 ((⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ ((𝑤𝑔𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
19 anass 684 . . . . . . . . . . . 12 (((𝑤𝑔𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2018, 19bitri 264 . . . . . . . . . . 11 ((⟨𝑤, 𝑔⟩ ∈ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2115, 16, 203bitri 286 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ (𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2221eubii 2629 . . . . . . . . 9 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23 euanv 2672 . . . . . . . . 9 (∃!𝑔(𝑤 ∧ (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ (𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
2422, 23bitr2i 265 . . . . . . . 8 ((𝑤 ∧ ∃!𝑔(𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵)
2514, 24syl6ib 241 . . . . . . 7 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤) → ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵))
26 euex 2631 . . . . . . . 8 (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔𝑤, 𝑔⟩ ∈ 𝐵)
27 nfeu1 2617 . . . . . . . . . 10 𝑔∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵
28 nfv 1992 . . . . . . . . . 10 𝑔(𝐵𝑤) ∈ 𝑤
2927, 28nfim 1974 . . . . . . . . 9 𝑔(∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)
3021simprbi 483 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝑔𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
3130simpld 477 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝐵𝑔𝑤)
32 tz6.12 6373 . . . . . . . . . . . . 13 ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵) → (𝐵𝑤) = 𝑔)
3332eleq1d 2824 . . . . . . . . . . . 12 ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵) → ((𝐵𝑤) ∈ 𝑤𝑔𝑤))
3433biimparc 505 . . . . . . . . . . 11 ((𝑔𝑤 ∧ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵)) → (𝐵𝑤) ∈ 𝑤)
3534exp32 632 . . . . . . . . . 10 (𝑔𝑤 → (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤)))
3631, 35mpcom 38 . . . . . . . . 9 (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵𝑤) ∈ 𝑤))
3729, 36exlimi 2233 . . . . . . . 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 453 . . . . 5 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → (𝑤 → (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4140ralrimiv 3103 . . . 4 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤))
42 vex 3343 . . . . . . 7 𝑦 ∈ V
4342inex2 4952 . . . . . 6 ( 𝐴𝑦) ∈ V
442, 43eqeltri 2835 . . . . 5 𝐵 ∈ V
45 fveq1 6352 . . . . . . . 8 (𝑓 = 𝐵 → (𝑓𝑤) = (𝐵𝑤))
4645eleq1d 2824 . . . . . . 7 (𝑓 = 𝐵 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝐵𝑤) ∈ 𝑤))
4746imbi2d 329 . . . . . 6 (𝑓 = 𝐵 → ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4847ralbidv 3124 . . . . 5 (𝑓 = 𝐵 → (∀𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ↔ ∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤)))
4944, 48spcev 3440 . . . 4 (∀𝑤 (𝑤 ≠ ∅ → (𝐵𝑤) ∈ 𝑤) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
5041, 49syl 17 . . 3 (∀𝑧𝐴 ∃!𝑣 𝑣 ∈ (𝑧𝑦) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
5150exlimiv 2007 . 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 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  {cab 2746  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cin 3714  c0 4058  {csn 4321  cop 4327   cuni 4588   × cxp 5264  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-iota 6012  df-fv 6057
This theorem is referenced by:  dfac5  9161
  Copyright terms: Public domain W3C validator