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Theorem dfac5lem3 9543
Description: Lemma for dfac5 9546. (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:   𝑤,𝑢,𝑡,   𝑤,𝐴
Allowed substitution hints:   𝐴(𝑢,𝑡,)

Proof of Theorem dfac5lem3
StepHypRef Expression
1 snex 5327 . . . 4 {𝑤} ∈ V
2 vex 3502 . . . 4 𝑤 ∈ V
31, 2xpex 7468 . . 3 ({𝑤} × 𝑤) ∈ V
4 neeq1 3082 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5 eqeq1 2829 . . . . 5 (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
65rexbidv 3301 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (∃𝑡 𝑢 = ({𝑡} × 𝑡) ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
74, 6anbi12d 630 . . 3 (𝑢 = ({𝑤} × 𝑤) → ((𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))))
83, 7elab 3670 . 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 2908 . 2 (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
11 xpeq2 5574 . . . . . 6 (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12 xp0 6012 . . . . . 6 ({𝑤} × ∅) = ∅
1311, 12syl6eq 2876 . . . . 5 (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14 rneq 5804 . . . . . 6 (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
152snnz 4709 . . . . . . 7 {𝑤} ≠ ∅
16 rnxp 6024 . . . . . . 7 ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
1715, 16ax-mp 5 . . . . . 6 ran ({𝑤} × 𝑤) = 𝑤
18 rn0 5794 . . . . . 6 ran ∅ = ∅
1914, 17, 183eqtr3g 2883 . . . . 5 (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
2013, 19impbii 210 . . . 4 (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
2120necon3bii 3072 . . 3 (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22 df-rex 3148 . . . 4 (∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23 rneq 5804 . . . . . . . . 9 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24 vex 3502 . . . . . . . . . . 11 𝑡 ∈ V
2524snnz 4709 . . . . . . . . . 10 {𝑡} ≠ ∅
26 rnxp 6024 . . . . . . . . . 10 ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
2725, 26ax-mp 5 . . . . . . . . 9 ran ({𝑡} × 𝑡) = 𝑡
2823, 17, 273eqtr3g 2883 . . . . . . . 8 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29 sneq 4573 . . . . . . . . . 10 (𝑤 = 𝑡 → {𝑤} = {𝑡})
3029xpeq1d 5582 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31 xpeq2 5574 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
3230, 31eqtrd 2860 . . . . . . . 8 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
3328, 32impbii 210 . . . . . . 7 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34 equcom 2018 . . . . . . 7 (𝑤 = 𝑡𝑡 = 𝑤)
3533, 34bitri 276 . . . . . 6 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑡 = 𝑤)
3635anbi1ci 625 . . . . 5 ((𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡 = 𝑤𝑡))
3736exbii 1841 . . . 4 (∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤𝑡))
38 elequ1 2114 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
3938equsexvw 2004 . . . 4 (∃𝑡(𝑡 = 𝑤𝑡) ↔ 𝑤)
4022, 37, 393bitrri 299 . . 3 (𝑤 ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
4121, 40anbi12i 626 . 2 ((𝑤 ≠ ∅ ∧ 𝑤) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
428, 10, 413bitr4i 304 1 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  {cab 2803  wne 3020  wrex 3143  c0 4294  {csn 4563   × cxp 5551  ran crn 5554
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564
This theorem is referenced by:  dfac5lem5  9545
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