Theorem dfac5lem3 10125

Description: Lemma for dfac5 10128. (Contributed by NM, 12-Apr-2004.)

Hypothesis

Ref	Expression
dfac5lem.1	⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}

Assertion

Ref	Expression
dfac5lem3	⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))

Distinct variable groups:   𝑡,ℎ,𝑢,𝑤   𝑤,𝐴

Allowed substitution hints:   𝐴(𝑢, 𝑡, ℎ)

Proof of Theorem dfac5lem3

Step	Hyp	Ref	Expression
1		vsnex 5408	. . . 4 ⊢ {𝑤} ∈ V
2		vex 3461	. . . 4 ⊢ 𝑤 ∈ V
3	1, 2	xpex 7758	. . 3 ⊢ ({𝑤} × 𝑤) ∈ V
4		neeq1 3022	. . . 4 ⊢ (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5		eqeq1 2769	. . . . 5 ⊢ (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
6	5	rexbidv 3191	. . . 4 ⊢ (𝑢 = ({𝑤} × 𝑤) → (∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡) ↔ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
7	4, 6	anbi12d 644	. . 3 ⊢ (𝑢 = ({𝑤} × 𝑤) → ((𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡)) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡))))
8	3, 7	elab 3640	. 2 ⊢ (({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
9		dfac5lem.1	. . 3 ⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
10	9	eleq2i 2857	. 2 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))})
11		xpeq2 5684	. . . . . 6 ⊢ (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12		xp0 5763	. . . . . 6 ⊢ ({𝑤} × ∅) = ∅
13	11, 12	eqtrdi 2816	. . . . 5 ⊢ (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14		rneq 5928	. . . . . 6 ⊢ (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
15	2	snnz 4744	. . . . . . 7 ⊢ {𝑤} ≠ ∅
16		rnxp 6170	. . . . . . 7 ⊢ ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
17	15, 16	ax-mp 5	. . . . . 6 ⊢ ran ({𝑤} × 𝑤) = 𝑤
18		rn0 5918	. . . . . 6 ⊢ ran ∅ = ∅
19	14, 17, 18	3eqtr3g 2823	. . . . 5 ⊢ (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
20	13, 19	impbii 212	. . . 4 ⊢ (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
21	20	necon3bii 3012	. . 3 ⊢ (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22		df-rex 3092	. . . 4 ⊢ (∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23		rneq 5928	. . . . . . . . 9 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24		vex 3461	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
25	24	snnz 4744	. . . . . . . . . 10 ⊢ {𝑡} ≠ ∅
26		rnxp 6170	. . . . . . . . . 10 ⊢ ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ ran ({𝑡} × 𝑡) = 𝑡
28	23, 17, 27	3eqtr3g 2823	. . . . . . . 8 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29		sneq 4601	. . . . . . . . . 10 ⊢ (𝑤 = 𝑡 → {𝑤} = {𝑡})
30	29	xpeq1d 5692	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31		xpeq2 5684	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
32	30, 31	eqtrd 2800	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
33	28, 32	impbii 212	. . . . . . 7 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34		equcom 2051	. . . . . . 7 ⊢ (𝑤 = 𝑡 ↔ 𝑡 = 𝑤)
35	33, 34	bitri 278	. . . . . 6 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑡 = 𝑤)
36	35	anbi1ci 638	. . . . 5 ⊢ ((𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ))
37	36	exbii 1881	. . . 4 ⊢ (∃𝑡(𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ))
38		elequ1 2153	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ ℎ ↔ 𝑤 ∈ ℎ))
39	38	equsexvw 2038	. . . 4 ⊢ (∃𝑡(𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ) ↔ 𝑤 ∈ ℎ)
40	22, 37, 39	3bitrri 301	. . 3 ⊢ (𝑤 ∈ ℎ ↔ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
41	21, 40	anbi12i 640	. 2 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
42	8, 10, 41	3bitr4i 306	1 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∃wrex 3091  ∅c0 4286  {csn 4591   × cxp 5661  ran crn 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  dfac5lem5  10127