Theorem dfac5lem3 10081

Description: Lemma for dfac5 10085. (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 3458	. . . 4 ⊢ 𝑤 ∈ V
3	1, 2	xpex 7736	. . 3 ⊢ ({𝑤} × 𝑤) ∈ V
4		neeq1 3019	. . . 4 ⊢ (𝑢 = ({𝑤} × 𝑤) → (𝑢 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅))
5		eqeq1 2766	. . . . 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 641	. . 3 ⊢ (𝑢 = ({𝑤} × 𝑤) → ((𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡)) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡))))
8	3, 7	elab 3638	. 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 2854	. 2 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ ({𝑤} × 𝑤) ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))})
11		xpeq2 5668	. . . . . 6 ⊢ (𝑤 = ∅ → ({𝑤} × 𝑤) = ({𝑤} × ∅))
12		xp0 5747	. . . . . 6 ⊢ ({𝑤} × ∅) = ∅
13	11, 12	eqtrdi 2813	. . . . 5 ⊢ (𝑤 = ∅ → ({𝑤} × 𝑤) = ∅)
14		rneq 5912	. . . . . 6 ⊢ (({𝑤} × 𝑤) = ∅ → ran ({𝑤} × 𝑤) = ran ∅)
15	2	snnz 4735	. . . . . . 7 ⊢ {𝑤} ≠ ∅
16		rnxp 6156	. . . . . . 7 ⊢ ({𝑤} ≠ ∅ → ran ({𝑤} × 𝑤) = 𝑤)
17	15, 16	ax-mp 5	. . . . . 6 ⊢ ran ({𝑤} × 𝑤) = 𝑤
18		rn0 5902	. . . . . 6 ⊢ ran ∅ = ∅
19	14, 17, 18	3eqtr3g 2820	. . . . 5 ⊢ (({𝑤} × 𝑤) = ∅ → 𝑤 = ∅)
20	13, 19	impbii 211	. . . 4 ⊢ (𝑤 = ∅ ↔ ({𝑤} × 𝑤) = ∅)
21	20	necon3bii 3009	. . 3 ⊢ (𝑤 ≠ ∅ ↔ ({𝑤} × 𝑤) ≠ ∅)
22		df-rex 3087	. . . 4 ⊢ (∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ ∃𝑡(𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
23		rneq 5912	. . . . . . . . 9 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → ran ({𝑤} × 𝑤) = ran ({𝑡} × 𝑡))
24		vex 3458	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
25	24	snnz 4735	. . . . . . . . . 10 ⊢ {𝑡} ≠ ∅
26		rnxp 6156	. . . . . . . . . 10 ⊢ ({𝑡} ≠ ∅ → ran ({𝑡} × 𝑡) = 𝑡)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ ran ({𝑡} × 𝑡) = 𝑡
28	23, 17, 27	3eqtr3g 2820	. . . . . . . 8 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) → 𝑤 = 𝑡)
29		sneq 4592	. . . . . . . . . 10 ⊢ (𝑤 = 𝑡 → {𝑤} = {𝑡})
30	29	xpeq1d 5676	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑤))
31		xpeq2 5668	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → ({𝑡} × 𝑤) = ({𝑡} × 𝑡))
32	30, 31	eqtrd 2797	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
33	28, 32	impbii 211	. . . . . . 7 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑤 = 𝑡)
34		equcom 2038	. . . . . . 7 ⊢ (𝑤 = 𝑡 ↔ 𝑡 = 𝑤)
35	33, 34	bitri 277	. . . . . 6 ⊢ (({𝑤} × 𝑤) = ({𝑡} × 𝑡) ↔ 𝑡 = 𝑤)
36	35	anbi1ci 635	. . . . 5 ⊢ ((𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ (𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ))
37	36	exbii 1868	. . . 4 ⊢ (∃𝑡(𝑡 ∈ ℎ ∧ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ))
38		elequ1 2149	. . . . 5 ⊢ (𝑡 = 𝑤 → (𝑡 ∈ ℎ ↔ 𝑤 ∈ ℎ))
39	38	equsexvw 2025	. . . 4 ⊢ (∃𝑡(𝑡 = 𝑤 ∧ 𝑡 ∈ ℎ) ↔ 𝑤 ∈ ℎ)
40	22, 37, 39	3bitrri 300	. . 3 ⊢ (𝑤 ∈ ℎ ↔ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡))
41	21, 40	anbi12i 637	. 2 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) ↔ (({𝑤} × 𝑤) ≠ ∅ ∧ ∃𝑡 ∈ ℎ ({𝑤} × 𝑤) = ({𝑡} × 𝑡)))
42	8, 10, 41	3bitr4i 305	1 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∃wrex 3086  ∅c0 4285  {csn 4582   × cxp 5645  ran crn 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658

This theorem is referenced by:  dfac5lem5  10083