Theorem dfac5lem3 10204

Description: Lemma for dfac5 10207. (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 5392	. . . 4 ⊢ {𝑤} ∈ V
2		vex 3454	. . . 4 ⊢ 𝑤 ∈ V
3	1, 2	xpex 7767	. . 3 ⊢ ({𝑤} × 𝑤) ∈ V
4		neeq1 3017	. . . 4 ⊢ (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5		eqeq1 2764	. . . . 5 ⊢ (𝑢 = ({𝑤} × 𝑤) → (𝑢 = ({𝑡} × 𝑡) ↔ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
6	5	rexbidv 3186	. . . 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 3632	. 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 2852	. 2 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))})
11		xpeq2 5671	. . . . . 6 ⊢ (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12		xp0 5750	. . . . . 6 ⊢ ({𝑤} × ∅) = ∅
13	11, 12	eqtrdi 2811	. . . . 5 ⊢ (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14		rneq 5917	. . . . . 6 ⊢ (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
15	2	snnz 4736	. . . . . . 7 ⊢ {𝑤} ≠ ∅
16		rnxp 6161	. . . . . . 7 ⊢ ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
17	15, 16	ax-mp 5	. . . . . 6 ⊢ ran ({𝑤} × 𝑤) = 𝑤
18		rn0 5907	. . . . . 6 ⊢ ran ∅ = ∅
19	14, 17, 18	3eqtr3g 2818	. . . . 5 ⊢ (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
20	13, 19	impbii 212	. . . 4 ⊢ (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
21	20	necon3bii 3007	. . 3 ⊢ (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22		df-rex 3087	. . . 4 ⊢ (∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23		rneq 5917	. . . . . . . . 9 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24		vex 3454	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
25	24	snnz 4736	. . . . . . . . . 10 ⊢ {𝑡} ≠ ∅
26		rnxp 6161	. . . . . . . . . 10 ⊢ ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ ran ({𝑡} × 𝑡) = 𝑡
28	23, 17, 27	3eqtr3g 2818	. . . . . . . 8 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29		sneq 4593	. . . . . . . . . 10 ⊢ (𝑤 = 𝑡 → {𝑤} = {𝑡})
30	29	xpeq1d 5679	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31		xpeq2 5671	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
32	30, 31	eqtrd 2795	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
33	28, 32	impbii 212	. . . . . . 7 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34		equcom 2050	. . . . . . 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 1880	. . . 4 ⊢ (∃𝑡(𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ))
38		elequ1 2151	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ ℎ ↔ 𝑤 ∈ ℎ))
39	38	equsexvw 2037	. . . 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 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∃wrex 3086  ∅c0 4278  {csn 4583   × cxp 5648  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by:  dfac5lem5  10206