Theorem dfac5lem5 10041

Description: Lemma for dfac5 10043. (Contributed by NM, 12-Apr-2004.)

Hypotheses

Ref	Expression
dfac5lem.1	⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
dfac5lem.2	⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
dfac5lem.3	⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦)

Assertion

Ref	Expression
dfac5lem5	⊢ (𝜑 → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))

Distinct variable groups:   𝑓,ℎ,𝑡,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐵,𝑓,𝑤,𝑧   𝑤,𝐴,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,ℎ)   𝐴(𝑣,𝑢,𝑡,𝑓,ℎ)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑡,ℎ)

Proof of Theorem dfac5lem5

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac5lem.1	. . 3 ⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
2		dfac5lem.2	. . 3 ⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
3	1, 2	dfac5lem4 10040	. 2 ⊢ (𝜑 → ∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
4		simpr 485	. . . . . . . . . 10 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ)
5	4	a1i 11	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ))
6		ineq1 4143	. . . . . . . . . . . . 13 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑧 ∩ 𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
7	6	eleq2d 2825	. . . . . . . . . . . 12 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
8	7	eubidv 2590	. . . . . . . . . . 11 ⊢ (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
9	8	rspccv 3557	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
10	1	dfac5lem3 10039	. . . . . . . . . 10 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))
11		dfac5lem1 10037	. . . . . . . . . 10 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
12	9, 10, 11	3imtr3g 296	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
13	5, 12	jcad 517	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
14		dfac5lem.3	. . . . . . . . . . . 12 ⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦)
15	14	eleq2i 2831	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦))
16		elin 3899	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
17	1	dfac5lem2 10038	. . . . . . . . . . . . 13 ⊢ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ↔ (𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤))
18	17	anbi1i 630	. . . . . . . . . . . 12 ⊢ ((⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ ((𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
19		anass 469	. . . . . . . . . . . 12 ⊢ (((𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
20	18, 19	bitri 276	. . . . . . . . . . 11 ⊢ ((⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
21	15, 16, 20	3bitri 298	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
22	21	eubii 2589	. . . . . . . . 9 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23		euanv 2628	. . . . . . . . 9 ⊢ (∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
24	22, 23	bitr2i 277	. . . . . . . 8 ⊢ ((𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
25	13, 24	imbitrdi 252	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵))
26		euex 2581	. . . . . . . 8 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
27		nfeu1 2593	. . . . . . . . . 10 ⊢ Ⅎ𝑔∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵
28		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝐵‘𝑤) ∈ 𝑤
29	27, 28	nfim 1903	. . . . . . . . 9 ⊢ Ⅎ𝑔(∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤)
30	21	simprbi 498	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
31	30	simpld 495	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → 𝑔 ∈ 𝑤)
32		tz6.12 6852	. . . . . . . . . . . . 13 ⊢ ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵) → (𝐵‘𝑤) = 𝑔)
33	32	eleq1d 2824	. . . . . . . . . . . 12 ⊢ ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵) → ((𝐵‘𝑤) ∈ 𝑤 ↔ 𝑔 ∈ 𝑤))
34	33	biimparc 480	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ 𝑤 ∧ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)) → (𝐵‘𝑤) ∈ 𝑤)
35	34	exp32 421	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑤 → (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤)))
36	31, 35	mpcom 38	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤))
37	29, 36	exlimi 2229	. . . . . . . 8 ⊢ (∃𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤))
38	26, 37	mpcom 38	. . . . . . 7 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤)
39	25, 38	syl6 35	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → (𝐵‘𝑤) ∈ 𝑤))
40	39	expcomd 417	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (𝑤 ∈ ℎ → (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
41	40	ralrimiv 3130	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤))
42		vex 3435	. . . . . . 7 ⊢ 𝑦 ∈ V
43	42	inex2 5247	. . . . . 6 ⊢ (∪ 𝐴 ∩ 𝑦) ∈ V
44	14, 43	eqeltri 2835	. . . . 5 ⊢ 𝐵 ∈ V
45		fveq1 6827	. . . . . . . 8 ⊢ (𝑓 = 𝐵 → (𝑓‘𝑤) = (𝐵‘𝑤))
46	45	eleq1d 2824	. . . . . . 7 ⊢ (𝑓 = 𝐵 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝐵‘𝑤) ∈ 𝑤))
47	46	imbi2d 341	. . . . . 6 ⊢ (𝑓 = 𝐵 → ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
48	47	ralbidv 3162	. . . . 5 ⊢ (𝑓 = 𝐵 → (∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
49	44, 48	spcev 3544	. . . 4 ⊢ (∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
50	41, 49	syl 17	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
51	50	exlimiv 1937	. 2 ⊢ (∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
52	3, 51	syl 17	1 ⊢ (𝜑 → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572  {cab 2717   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∩ cin 3882  ∅c0 4262  {csn 4556  ⟨cop 4562  ∪ cuni 4839   × cxp 5617  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-iota 6442  df-fv 6494

This theorem is referenced by:  dfac5  10043