Theorem ptcmplem1 23939

Description: Lemma for ptcmp 23945. (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 6689	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		eqid 2729	. . . . . . . 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 23457	. . . . . . 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 584	. . . . . 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 23282	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Comp → 𝑥 ∈ Top)
8	7	ssriv 3950	. . . . . . . . . 10 ⊢ Comp ⊆ Top
9		fss 6704	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
10	2, 8, 9	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶Top)
11		ptcmp.2	. . . . . . . . . 10 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
12	4, 11	ptbasfi 23468	. . . . . . . . 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 584	. . . . . . . 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 4121	. . . . . . . . . 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 5901	. . . . . . . . . . 11 ⊢ ran 𝑆 = ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
17	16	uneq2i 4128	. . . . . . . . . 10 ⊢ ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
18	14, 17	eqtri 2752	. . . . . . . . 9 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
19	18	fveq2i 6861	. . . . . . . 8 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
20	13, 19	eqtr4di 2782	. . . . . . 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 6862	. . . . . 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 2764	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
23	22	unieqd 4884	. . . 4 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24		fibas 22864	. . . . 5 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25		unitg 22854	. . . . 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 2780	. . 3 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
28		eqid 2729	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
29	28	ptuni 23481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
30	1, 10, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
31	11, 30	eqtrid 2776	. . 3 ⊢ (𝜑 → 𝑋 = ∪ (∏t‘𝐹))
32		ptcmp.5	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
33	32	pwexd 5334	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
34		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
35	34	mptpreima 6211	. . . . . . . . . . 11 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
36	35	ssrab3 4045	. . . . . . . . . 10 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋
37	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38		elpw2g 5288	. . . . . . . . . . 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 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
41	40	ralrimivva 3180	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
42	15	fmpox 8046	. . . . . . . 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 218	. . . . . . 7 ⊢ (𝜑 → 𝑆:∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝒫 𝑋)
44	43	frnd 6696	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
45	33, 44	ssexd 5279	. . . . 5 ⊢ (𝜑 → ran 𝑆 ∈ V)
46		snex 5391	. . . . 5 ⊢ {𝑋} ∈ V
47		unexg 7719	. . . . 5 ⊢ ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
48	45, 46, 47	sylancl 586	. . . 4 ⊢ (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49		fiuni 9379	. . . 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 2774	. 2 ⊢ (𝜑 → 𝑋 = ∪ (ran 𝑆 ∪ {𝑋}))
52	51, 22	jca 511	1 ⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511   ∈ cmpo 7389  Xcixp 8870  Fincfn 8918  ficfi 9361  cardccrd 9888  topGenctg 17400  ∏_tcpt 17401  Topctop 22780  TopBasesctb 22832  Compccmp 23273  UFLcufl 23787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-2o 8435  df-ixp 8871  df-en 8919  df-dom 8920  df-fin 8922  df-fi 9362  df-topgen 17406  df-pt 17407  df-top 22781  df-bases 22833  df-cmp 23274

This theorem is referenced by:  ptcmplem5  23943