Theorem ptcmplem1 24220

Description: Lemma for ptcmp 24226. (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 6706	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		eqid 2762	. . . . . . . 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 23738	. . . . . . 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 595	. . . . . 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 23563	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Comp → 𝑥 ∈ Top)
8	7	ssriv 3940	. . . . . . . . . 10 ⊢ Comp ⊆ Top
9		fss 6722	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
10	2, 8, 9	sylancl 597	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶Top)
11		ptcmp.2	. . . . . . . . . 10 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
12	4, 11	ptbasfi 23749	. . . . . . . . 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 595	. . . . . . . 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 4111	. . . . . . . . . 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 5926	. . . . . . . . . . 11 ⊢ ran 𝑆 = ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
17	16	uneq2i 4118	. . . . . . . . . 10 ⊢ ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
18	14, 17	eqtri 2785	. . . . . . . . 9 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
19	18	fveq2i 6884	. . . . . . . 8 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
20	13, 19	eqtr4di 2815	. . . . . . 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 6885	. . . . . 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 2797	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
23	22	unieqd 4884	. . . 4 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24		fibas 23145	. . . . 5 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25		unitg 23135	. . . . 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 2813	. . 3 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
28		eqid 2762	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
29	28	ptuni 23762	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
30	1, 10, 29	syl2anc 595	. . . 4 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
31	11, 30	eqtrid 2809	. . 3 ⊢ (𝜑 → 𝑋 = ∪ (∏t‘𝐹))
32		ptcmp.5	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
33	32	pwexd 5349	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
34		eqid 2762	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
35	34	mptpreima 6238	. . . . . . . . . . 11 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
36	35	ssrab3 4035	. . . . . . . . . 10 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋
37	32	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38		elpw2g 5303	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (UFL ∩ dom card) → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ⊆ 𝑋))
39	37, 38	syl 18	. . . . . . . . . 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 3207	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
42	15	fmpox 8062	. . . . . . . 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 6714	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
45	33, 44	ssexd 5294	. . . . 5 ⊢ (𝜑 → ran 𝑆 ∈ V)
46		snex 5409	. . . . 5 ⊢ {𝑋} ∈ V
47		unexg 7743	. . . . 5 ⊢ ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
48	45, 46, 47	sylancl 597	. . . 4 ⊢ (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49		fiuni 9386	. . . 4 ⊢ ((ran 𝑆 ∪ {𝑋}) ∈ V → ∪ (ran 𝑆 ∪ {𝑋}) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
50	48, 49	syl 18	. . 3 ⊢ (𝜑 → ∪ (ran 𝑆 ∪ {𝑋}) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
51	27, 31, 50	3eqtr4d 2807	. 2 ⊢ (𝜑 → 𝑋 = ∪ (ran 𝑆 ∪ {𝑋}))
52	51, 22	jca 520	1 ⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191   × cxp 5658  ^◡ccnv 5659  dom cdm 5660  ran crn 5661   “ cima 5663   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536   ∈ cmpo 7414  Xcixp 8893  Fincfn 8941  ficfi 9368  cardccrd 9928  topGenctg 17496  ∏_tcpt 17497  Topctop 23061  TopBasesctb 23113  Compccmp 23554  UFLcufl 24068

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-1o 8451  df-2o 8452  df-ixp 8894  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9369  df-topgen 17502  df-pt 17503  df-top 23062  df-bases 23114  df-cmp 23555

This theorem is used by:  ptcmplem5  24224