Theorem ptcmplem1 22949

Description: Lemma for ptcmp 22955. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
ptcmp.1	⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
ptcmp.2	⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
ptcmp.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ptcmp.4	⊢ (𝜑 → 𝐹:𝐴⟶Comp)
ptcmp.5	⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))

Assertion

Ref	Expression
ptcmplem1	⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))

Distinct variable groups:   𝑘,𝑛,𝑢,𝑤,𝐴   𝑆,𝑘,𝑛,𝑢   𝜑,𝑘,𝑛,𝑢   𝑘,𝑉,𝑛,𝑢,𝑤   𝑘,𝐹,𝑛,𝑢,𝑤   𝑘,𝑋,𝑛,𝑢,𝑤

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)

Proof of Theorem ptcmplem1

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

Step	Hyp	Ref	Expression
1		ptcmp.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		ptcmp.4	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶Comp)
3	2	ffnd 6546	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		eqid 2737	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
5	4	ptval 22467	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
6	1, 3, 5	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
7		cmptop 22292	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Comp → 𝑥 ∈ Top)
8	7	ssriv 3905	. . . . . . . . . 10 ⊢ Comp ⊆ Top
9		fss 6562	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
10	2, 8, 9	sylancl 589	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶Top)
11		ptcmp.2	. . . . . . . . . 10 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
12	4, 11	ptbasfi 22478	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
13	1, 10, 12	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
14		uncom 4067	. . . . . . . . . 10 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
15		ptcmp.1	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
16	15	rneqi 5806	. . . . . . . . . . 11 ⊢ ran 𝑆 = ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
17	16	uneq2i 4074	. . . . . . . . . 10 ⊢ ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
18	14, 17	eqtri 2765	. . . . . . . . 9 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
19	18	fveq2i 6720	. . . . . . . 8 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
20	13, 19	eqtr4di 2796	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = (fi‘(ran 𝑆 ∪ {𝑋})))
21	20	fveq2d 6721	. . . . . 6 ⊢ (𝜑 → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
22	6, 21	eqtrd 2777	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
23	22	unieqd 4833	. . . 4 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24		fibas 21874	. . . . 5 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25		unitg 21864	. . . . 5 ⊢ ((fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases → ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
26	24, 25	ax-mp 5	. . . 4 ⊢ ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = ∪ (fi‘(ran 𝑆 ∪ {𝑋}))
27	23, 26	eqtrdi 2794	. . 3 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
28		eqid 2737	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
29	28	ptuni 22491	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
30	1, 10, 29	syl2anc 587	. . . 4 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
31	11, 30	syl5eq 2790	. . 3 ⊢ (𝜑 → 𝑋 = ∪ (∏t‘𝐹))
32		ptcmp.5	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
33	32	pwexd 5272	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
34		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
35	34	mptpreima 6101	. . . . . . . . . . 11 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
36	35	ssrab3 3995	. . . . . . . . . 10 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋
37	32	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38		elpw2g 5237	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (UFL ∩ dom card) → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋))
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋))
40	36, 39	mpbiri 261	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
41	40	ralrimivva 3112	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
42	15	fmpox 7837	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ 𝑆:∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝒫 𝑋)
43	41, 42	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑆:∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝒫 𝑋)
44	43	frnd 6553	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
45	33, 44	ssexd 5217	. . . . 5 ⊢ (𝜑 → ran 𝑆 ∈ V)
46		snex 5324	. . . . 5 ⊢ {𝑋} ∈ V
47		unexg 7534	. . . . 5 ⊢ ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
48	45, 46, 47	sylancl 589	. . . 4 ⊢ (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49		fiuni 9044	. . . 4 ⊢ ((ran 𝑆 ∪ {𝑋}) ∈ V → ∪ (ran 𝑆 ∪ {𝑋}) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → ∪ (ran 𝑆 ∪ {𝑋}) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
51	27, 31, 50	3eqtr4d 2787	. 2 ⊢ (𝜑 → 𝑋 = ∪ (ran 𝑆 ∪ {𝑋}))
52	51, 22	jca 515	1 ⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2110  {cab 2714  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  𝒫 cpw 4513  {csn 4541  ∪ cuni 4819  ∪ ciun 4904   ↦ cmpt 5135   × cxp 5549  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   “ cima 5554   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380   ∈ cmpo 7215  Xcixp 8578  Fincfn 8626  ficfi 9026  cardccrd 9551  topGenctg 16942  ∏_tcpt 16943  Topctop 21790  TopBasesctb 21842  Compccmp 22283  UFLcufl 22797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-1o 8202  df-er 8391  df-ixp 8579  df-en 8627  df-dom 8628  df-fin 8630  df-fi 9027  df-topgen 16948  df-pt 16949  df-top 21791  df-bases 21843  df-cmp 22284

This theorem is referenced by:  ptcmplem5  22953