Theorem dfac21 43055

Description: Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)

Assertion

Ref	Expression
dfac21	⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))

Proof of Theorem dfac21

Dummy variables 𝑔 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3482	. . . . . . 7 ⊢ 𝑓 ∈ V
2	1	dmex 7932	. . . . . 6 ⊢ dom 𝑓 ∈ V
3	2	a1i 11	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4		simpr 484	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5		fvex 6920	. . . . . . . 8 ⊢ (∏t‘𝑓) ∈ V
6	5	uniex 7760	. . . . . . 7 ⊢ ∪ (∏t‘𝑓) ∈ V
7		acufl 23941	. . . . . . . 8 ⊢ (CHOICE → UFL = V)
8	7	adantr 480	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → UFL = V)
9	6, 8	eleqtrrid 2846	. . . . . 6 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ UFL)
10		simpl 482	. . . . . . . 8 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → CHOICE)
11		dfac10 10176	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
12	10, 11	sylib 218	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom card = V)
13	6, 12	eleqtrrid 2846	. . . . . 6 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ dom card)
14	9, 13	elind 4210	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card))
15		eqid 2735	. . . . . 6 ⊢ (∏t‘𝑓) = (∏t‘𝑓)
16		eqid 2735	. . . . . 6 ⊢ ∪ (∏t‘𝑓) = ∪ (∏t‘𝑓)
17	15, 16	ptcmpg 24081	. . . . 5 ⊢ ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card)) → (∏t‘𝑓) ∈ Comp)
18	3, 4, 14, 17	syl3anc 1370	. . . 4 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → (∏t‘𝑓) ∈ Comp)
19	18	ex 412	. . 3 ⊢ (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
20	19	alrimiv 1925	. 2 ⊢ (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
21		fvex 6920	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
22		kelac2lem 43053	. . . . . . . . 9 ⊢ ((𝑔‘𝑦) ∈ V → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) ∈ Comp)
23	21, 22	mp1i 13	. . . . . . . 8 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) ∈ Comp)
24	23	fmpttd 7135	. . . . . . 7 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom 𝑔⟶Comp)
25	24	ffdmd 6767	. . . . . 6 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp)
26		vex 3482	. . . . . . . . 9 ⊢ 𝑔 ∈ V
27	26	dmex 7932	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
28	27	mptex 7243	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) ∈ V
29		id 22	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})))
30		dmeq 5917	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})))
31	29, 30	feq12d 6725	. . . . . . . 8 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → (𝑓:dom 𝑓⟶Comp ↔ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp))
32		fveq2 6907	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → (∏t‘𝑓) = (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))))
33	32	eleq1d 2824	. . . . . . . 8 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → ((∏t‘𝑓) ∈ Comp ↔ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp))
34	31, 33	imbi12d 344	. . . . . . 7 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → ((𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) ↔ ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp)))
35	28, 34	spcv 3605	. . . . . 6 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp))
36	25, 35	syl5com 31	. . . . 5 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp))
37		fvex 6920	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
38	37	a1i 11	. . . . . . 7 ⊢ ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ V)
39		df-nel 3045	. . . . . . . . . . 11 ⊢ (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
40	39	biimpi 216	. . . . . . . . . 10 ⊢ (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
41	40	ad2antlr 727	. . . . . . . . 9 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
42		fvelrn 7096	. . . . . . . . . . . 12 ⊢ ((Fun 𝑔 ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
43	42	adantlr 715	. . . . . . . . . . 11 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
44		eleq1 2827	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
45	43, 44	syl5ibcom 245	. . . . . . . . . 10 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔‘𝑥) = ∅ → ∅ ∈ ran 𝑔))
46	45	necon3bd 2952	. . . . . . . . 9 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔‘𝑥) ≠ ∅))
47	41, 46	mpd 15	. . . . . . . 8 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
48	47	adantlr 715	. . . . . . 7 ⊢ ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
49		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
50	49	unieqd 4925	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ∪ (𝑔‘𝑦) = ∪ (𝑔‘𝑥))
51	50	pweqd 4622	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → 𝒫 ∪ (𝑔‘𝑦) = 𝒫 ∪ (𝑔‘𝑥))
52	51	sneqd 4643	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → {𝒫 ∪ (𝑔‘𝑦)} = {𝒫 ∪ (𝑔‘𝑥)})
53	49, 52	preq12d 4746	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}} = {(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}})
54	53	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) = (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))
55	54	cbvmptv 5261	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))
56	55	fveq2i 6910	. . . . . . . . . 10 ⊢ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}})))
57	56	eleq1i 2830	. . . . . . . . 9 ⊢ ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp ↔ (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))) ∈ Comp)
58	57	biimpi 216	. . . . . . . 8 ⊢ ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))) ∈ Comp)
59	58	adantl 481	. . . . . . 7 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))) ∈ Comp)
60	38, 48, 59	kelac2 43054	. . . . . 6 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)
61	60	ex 412	. . . . 5 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
62	36, 61	syldc 48	. . . 4 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
63	62	alrimiv 1925	. . 3 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
64		dfac9 10175	. . 3 ⊢ (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
65	63, 64	sylibr 234	. 2 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → CHOICE)
66	20, 65	impbii 209	1 ⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∉ wnel 3044  Vcvv 3478   ∩ cin 3962  ∅c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633  ∪ cuni 4912   ↦ cmpt 5231  dom cdm 5689  ran crn 5690  Fun wfun 6557  ⟶wf 6559  ‘cfv 6563  Xcixp 8936  cardccrd 9973  CHOICEwac 10153  topGenctg 17484  ∏_tcpt 17485  Compccmp 23410  UFLcufl 23924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-wdom 9603  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-topgen 17490  df-pt 17491  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cmp 23411  df-fil 23870  df-ufil 23925  df-ufl 23926  df-flim 23963  df-fcls 23965

This theorem is referenced by: (None)