Theorem ptcmplem1 21666

Description: Lemma for ptcmp 21672. (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		ffn 5958	. . . . . . . 8 ⊢ (𝐹:𝐴⟶Comp → 𝐹 Fn 𝐴)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		eqid 2610	. . . . . . . 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))}
6	5	ptval 21183	. . . . . . 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))}))
7	1, 4, 6	syl2anc 691	. . . . . 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))}))
8		cmptop 21008	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Comp → 𝑥 ∈ Top)
9	8	ssriv 3572	. . . . . . . . . 10 ⊢ Comp ⊆ Top
10		fss 5969	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
11	2, 9, 10	sylancl 693	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶Top)
12		ptcmp.2	. . . . . . . . . 10 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
13	5, 12	ptbasfi 21194	. . . . . . . . 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)))))
14	1, 11, 13	syl2anc 691	. . . . . . . 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)))))
15		uncom 3719	. . . . . . . . . 10 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
16		ptcmp.1	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
17	16	rneqi 5273	. . . . . . . . . . 11 ⊢ ran 𝑆 = ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
18	17	uneq2i 3726	. . . . . . . . . 10 ⊢ ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
19	15, 18	eqtri 2632	. . . . . . . . 9 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
20	19	fveq2i 6106	. . . . . . . 8 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
21	14, 20	syl6eqr 2662	. . . . . . 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})))
22	21	fveq2d 6107	. . . . . 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}))))
23	7, 22	eqtrd 2644	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24	23	unieqd 4382	. . . 4 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
25		fibas 20592	. . . . 5 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
26		unitg 20582	. . . . 5 ⊢ ((fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases → ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
27	25, 26	ax-mp 5	. . . 4 ⊢ ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = ∪ (fi‘(ran 𝑆 ∪ {𝑋}))
28	24, 27	syl6eq 2660	. . 3 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
29		eqid 2610	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
30	29	ptuni 21207	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
31	1, 11, 30	syl2anc 691	. . . 4 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
32	12, 31	syl5eq 2656	. . 3 ⊢ (𝜑 → 𝑋 = ∪ (∏t‘𝐹))
33		ptcmp.5	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
34		pwexg 4776	. . . . . . 7 ⊢ (𝑋 ∈ (UFL ∩ dom card) → 𝒫 𝑋 ∈ V)
35	33, 34	syl 17	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
36		eqid 2610	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
37	36	mptpreima 5545	. . . . . . . . . . 11 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
38		ssrab2 3650	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢} ⊆ 𝑋
39	37, 38	eqsstri 3598	. . . . . . . . . 10 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋
40	33	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
41		elpw2g 4754	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (UFL ∩ dom card) → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋))
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋))
43	39, 42	mpbiri 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
44	43	ralrimivva 2954	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
45	16	fmpt2x 7125	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ 𝑆:∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝒫 𝑋)
46	44, 45	sylib 207	. . . . . . 7 ⊢ (𝜑 → 𝑆:∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝒫 𝑋)
47		frn 5966	. . . . . . 7 ⊢ (𝑆:∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝒫 𝑋 → ran 𝑆 ⊆ 𝒫 𝑋)
48	46, 47	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
49	35, 48	ssexd 4733	. . . . 5 ⊢ (𝜑 → ran 𝑆 ∈ V)
50		snex 4835	. . . . 5 ⊢ {𝑋} ∈ V
51		unexg 6857	. . . . 5 ⊢ ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
52	49, 50, 51	sylancl 693	. . . 4 ⊢ (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
53		fiuni 8217	. . . 4 ⊢ ((ran 𝑆 ∪ {𝑋}) ∈ V → ∪ (ran 𝑆 ∪ {𝑋}) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
54	52, 53	syl 17	. . 3 ⊢ (𝜑 → ∪ (ran 𝑆 ∪ {𝑋}) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
55	28, 32, 54	3eqtr4d 2654	. 2 ⊢ (𝜑 → 𝑋 = ∪ (ran 𝑆 ∪ {𝑋}))
56	55, 23	jca 553	1 ⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804   ↦ cmpt2 6551  Xcixp 7794  Fincfn 7841  ficfi 8199  cardccrd 8644  topGenctg 15921  ∏_tcpt 15922  Topctop 20517  TopBasesctb 20520  Compccmp 20999  UFLcufl 21514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ixp 7795  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-cmp 21000

This theorem is referenced by:  ptcmplem5  21670