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Theorem dfac5lem3 10163
Description: Lemma for dfac5 10167. (Contributed by NM, 12-Apr-2004.)
Hypothesis
Ref Expression
dfac5lem.1 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
Assertion
Ref Expression
dfac5lem3 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
Distinct variable groups:   𝑡,,𝑢,𝑤   𝑤,𝐴
Allowed substitution hints:   𝐴(𝑢,𝑡,)

Proof of Theorem dfac5lem3
StepHypRef Expression
1 vsnex 5440 . . . 4 {𝑤} ∈ V
2 vex 3482 . . . 4 𝑤 ∈ V
31, 2xpex 7772 . . 3 ({𝑤} × 𝑤) ∈ V
4 neeq1 3001 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5 eqeq1 2739 . . . . 5 (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
65rexbidv 3177 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (∃𝑡 𝑢 = ({𝑡} × 𝑡) ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
74, 6anbi12d 632 . . 3 (𝑢 = ({𝑤} × 𝑤) → ((𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))))
83, 7elab 3681 . 2 (({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
9 dfac5lem.1 . . 3 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
109eleq2i 2831 . 2 (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
11 xpeq2 5710 . . . . . 6 (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12 xp0 6180 . . . . . 6 ({𝑤} × ∅) = ∅
1311, 12eqtrdi 2791 . . . . 5 (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14 rneq 5950 . . . . . 6 (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
152snnz 4781 . . . . . . 7 {𝑤} ≠ ∅
16 rnxp 6192 . . . . . . 7 ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
1715, 16ax-mp 5 . . . . . 6 ran ({𝑤} × 𝑤) = 𝑤
18 rn0 5939 . . . . . 6 ran ∅ = ∅
1914, 17, 183eqtr3g 2798 . . . . 5 (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
2013, 19impbii 209 . . . 4 (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
2120necon3bii 2991 . . 3 (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22 df-rex 3069 . . . 4 (∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23 rneq 5950 . . . . . . . . 9 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24 vex 3482 . . . . . . . . . . 11 𝑡 ∈ V
2524snnz 4781 . . . . . . . . . 10 {𝑡} ≠ ∅
26 rnxp 6192 . . . . . . . . . 10 ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
2725, 26ax-mp 5 . . . . . . . . 9 ran ({𝑡} × 𝑡) = 𝑡
2823, 17, 273eqtr3g 2798 . . . . . . . 8 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29 sneq 4641 . . . . . . . . . 10 (𝑤 = 𝑡 → {𝑤} = {𝑡})
3029xpeq1d 5718 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31 xpeq2 5710 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
3230, 31eqtrd 2775 . . . . . . . 8 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
3328, 32impbii 209 . . . . . . 7 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34 equcom 2015 . . . . . . 7 (𝑤 = 𝑡𝑡 = 𝑤)
3533, 34bitri 275 . . . . . 6 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑡 = 𝑤)
3635anbi1ci 626 . . . . 5 ((𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡 = 𝑤𝑡))
3736exbii 1845 . . . 4 (∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤𝑡))
38 elequ1 2113 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
3938equsexvw 2002 . . . 4 (∃𝑡(𝑡 = 𝑤𝑡) ↔ 𝑤)
4022, 37, 393bitrri 298 . . 3 (𝑤 ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
4121, 40anbi12i 628 . 2 ((𝑤 ≠ ∅ ∧ 𝑤) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
428, 10, 413bitr4i 303 1 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wrex 3068  c0 4339  {csn 4631   × cxp 5687  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  dfac5lem5  10165
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