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Theorem dfac5lem3 8892
Description: Lemma for dfac5 8895. (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 4869 . . . 4 {𝑤} ∈ V
2 vex 3189 . . . 4 𝑤 ∈ V
31, 2xpex 6915 . . 3 ({𝑤} × 𝑤) ∈ V
4 neeq1 2852 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5 eqeq1 2625 . . . . 5 (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
65rexbidv 3045 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (∃𝑡 𝑢 = ({𝑡} × 𝑡) ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
74, 6anbi12d 746 . . 3 (𝑢 = ({𝑤} × 𝑤) → ((𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))))
83, 7elab 3333 . 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 2690 . 2 (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
11 xpeq2 5089 . . . . . 6 (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12 xp0 5511 . . . . . 6 ({𝑤} × ∅) = ∅
1311, 12syl6eq 2671 . . . . 5 (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14 rneq 5311 . . . . . 6 (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
152snnz 4279 . . . . . . 7 {𝑤} ≠ ∅
16 rnxp 5523 . . . . . . 7 ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
1715, 16ax-mp 5 . . . . . 6 ran ({𝑤} × 𝑤) = 𝑤
18 rn0 5337 . . . . . 6 ran ∅ = ∅
1914, 17, 183eqtr3g 2678 . . . . 5 (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
2013, 19impbii 199 . . . 4 (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
2120necon3bii 2842 . . 3 (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22 df-rex 2913 . . . 4 (∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23 rneq 5311 . . . . . . . . . 10 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24 vex 3189 . . . . . . . . . . . 12 𝑡 ∈ V
2524snnz 4279 . . . . . . . . . . 11 {𝑡} ≠ ∅
26 rnxp 5523 . . . . . . . . . . 11 ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
2725, 26ax-mp 5 . . . . . . . . . 10 ran ({𝑡} × 𝑡) = 𝑡
2823, 17, 273eqtr3g 2678 . . . . . . . . 9 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29 sneq 4158 . . . . . . . . . . 11 (𝑤 = 𝑡 → {𝑤} = {𝑡})
3029xpeq1d 5098 . . . . . . . . . 10 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31 xpeq2 5089 . . . . . . . . . 10 (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
3230, 31eqtrd 2655 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
3328, 32impbii 199 . . . . . . . 8 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34 equcom 1942 . . . . . . . 8 (𝑤 = 𝑡𝑡 = 𝑤)
3533, 34bitri 264 . . . . . . 7 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑡 = 𝑤)
3635anbi2i 729 . . . . . 6 ((𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡𝑡 = 𝑤))
37 ancom 466 . . . . . 6 ((𝑡𝑡 = 𝑤) ↔ (𝑡 = 𝑤𝑡))
3836, 37bitri 264 . . . . 5 ((𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡 = 𝑤𝑡))
3938exbii 1771 . . . 4 (∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤𝑡))
40 elequ1 1994 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
412, 40ceqsexv 3228 . . . 4 (∃𝑡(𝑡 = 𝑤𝑡) ↔ 𝑤)
4222, 39, 413bitrri 287 . . 3 (𝑤 ↔ ∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
4321, 42anbi12i 732 . 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 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wrex 2908  c0 3891  {csn 4148   × cxp 5072  ran crn 5075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085
This theorem is referenced by:  dfac5lem5  8894
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