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Theorem dfac5lem3 9134
 Description: Lemma for dfac5 9137. (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 5053 . . . 4 {𝑤} ∈ V
2 vex 3339 . . . 4 𝑤 ∈ V
31, 2xpex 7123 . . 3 ({𝑤} × 𝑤) ∈ V
4 neeq1 2990 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5 eqeq1 2760 . . . . 5 (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
65rexbidv 3186 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (∃𝑡 𝑢 = ({𝑡} × 𝑡) ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
74, 6anbi12d 749 . . 3 (𝑢 = ({𝑤} × 𝑤) → ((𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))))
83, 7elab 3486 . 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 2827 . 2 (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
11 xpeq2 5282 . . . . . 6 (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12 xp0 5706 . . . . . 6 ({𝑤} × ∅) = ∅
1311, 12syl6eq 2806 . . . . 5 (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14 rneq 5502 . . . . . 6 (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
152snnz 4448 . . . . . . 7 {𝑤} ≠ ∅
16 rnxp 5718 . . . . . . 7 ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
1715, 16ax-mp 5 . . . . . 6 ran ({𝑤} × 𝑤) = 𝑤
18 rn0 5528 . . . . . 6 ran ∅ = ∅
1914, 17, 183eqtr3g 2813 . . . . 5 (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
2013, 19impbii 199 . . . 4 (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
2120necon3bii 2980 . . 3 (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22 df-rex 3052 . . . 4 (∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23 rneq 5502 . . . . . . . . . 10 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24 vex 3339 . . . . . . . . . . . 12 𝑡 ∈ V
2524snnz 4448 . . . . . . . . . . 11 {𝑡} ≠ ∅
26 rnxp 5718 . . . . . . . . . . 11 ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
2725, 26ax-mp 5 . . . . . . . . . 10 ran ({𝑡} × 𝑡) = 𝑡
2823, 17, 273eqtr3g 2813 . . . . . . . . 9 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29 sneq 4327 . . . . . . . . . . 11 (𝑤 = 𝑡 → {𝑤} = {𝑡})
3029xpeq1d 5291 . . . . . . . . . 10 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31 xpeq2 5282 . . . . . . . . . 10 (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
3230, 31eqtrd 2790 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
3328, 32impbii 199 . . . . . . . 8 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34 equcom 2096 . . . . . . . 8 (𝑤 = 𝑡𝑡 = 𝑤)
3533, 34bitri 264 . . . . . . 7 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑡 = 𝑤)
3635anbi2i 732 . . . . . 6 ((𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡𝑡 = 𝑤))
37 ancom 465 . . . . . 6 ((𝑡𝑡 = 𝑤) ↔ (𝑡 = 𝑤𝑡))
3836, 37bitri 264 . . . . 5 ((𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡 = 𝑤𝑡))
3938exbii 1919 . . . 4 (∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤𝑡))
40 elequ1 2142 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
412, 40ceqsexv 3378 . . . 4 (∃𝑡(𝑡 = 𝑤𝑡) ↔ 𝑤)
4222, 39, 413bitrri 287 . . 3 (𝑤 ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
4321, 42anbi12i 735 . 2 ((𝑤 ≠ ∅ ∧ 𝑤) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
448, 10, 433bitr4i 292 1 (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1628  ∃wex 1849   ∈ wcel 2135  {cab 2742   ≠ wne 2928  ∃wrex 3047  ∅c0 4054  {csn 4317   × cxp 5260  ran crn 5263 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-xp 5268  df-rel 5269  df-cnv 5270  df-dm 5272  df-rn 5273 This theorem is referenced by:  dfac5lem5  9136
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