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Theorem dfac5lem3 9061
Description: Lemma for dfac5 9064. (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 5013 . . . 4 {𝑤} ∈ V
2 vex 3307 . . . 4 𝑤 ∈ V
31, 2xpex 7079 . . 3 ({𝑤} × 𝑤) ∈ V
4 neeq1 2958 . . . 4 (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5 eqeq1 2728 . . . . 5 (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
65rexbidv 3154 . . . 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 3455 . 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 2795 . 2 (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
11 xpeq2 5238 . . . . . 6 (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12 xp0 5662 . . . . . 6 ({𝑤} × ∅) = ∅
1311, 12syl6eq 2774 . . . . 5 (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14 rneq 5458 . . . . . 6 (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
152snnz 4415 . . . . . . 7 {𝑤} ≠ ∅
16 rnxp 5674 . . . . . . 7 ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
1715, 16ax-mp 5 . . . . . 6 ran ({𝑤} × 𝑤) = 𝑤
18 rn0 5484 . . . . . 6 ran ∅ = ∅
1914, 17, 183eqtr3g 2781 . . . . 5 (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
2013, 19impbii 199 . . . 4 (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
2120necon3bii 2948 . . 3 (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22 df-rex 3020 . . . 4 (∃𝑡 ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23 rneq 5458 . . . . . . . . . 10 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24 vex 3307 . . . . . . . . . . . 12 𝑡 ∈ V
2524snnz 4415 . . . . . . . . . . 11 {𝑡} ≠ ∅
26 rnxp 5674 . . . . . . . . . . 11 ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
2725, 26ax-mp 5 . . . . . . . . . 10 ran ({𝑡} × 𝑡) = 𝑡
2823, 17, 273eqtr3g 2781 . . . . . . . . 9 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29 sneq 4295 . . . . . . . . . . 11 (𝑤 = 𝑡 → {𝑤} = {𝑡})
3029xpeq1d 5247 . . . . . . . . . 10 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31 xpeq2 5238 . . . . . . . . . 10 (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
3230, 31eqtrd 2758 . . . . . . . . 9 (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
3328, 32impbii 199 . . . . . . . 8 (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34 equcom 2064 . . . . . . . 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 1887 . . . 4 (∃𝑡(𝑡 ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤𝑡))
40 elequ1 2110 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
412, 40ceqsexv 3346 . . . 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 1596  wex 1817  wcel 2103  {cab 2710  wne 2896  wrex 3015  c0 4023  {csn 4285   × cxp 5216  ran crn 5219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229
This theorem is referenced by:  dfac5lem5  9063
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