Theorem ptcmplem1 24000

Description: Lemma for ptcmp 24006. (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 6664	. . . . . . 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 23518	. . . . . . 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 585	. . . . . 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 23343	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Comp → 𝑥 ∈ Top)
8	7	ssriv 3938	. . . . . . . . . 10 ⊢ Comp ⊆ Top
9		fss 6679	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
10	2, 8, 9	sylancl 587	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶Top)
11		ptcmp.2	. . . . . . . . . 10 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
12	4, 11	ptbasfi 23529	. . . . . . . . 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 585	. . . . . . . 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 5887	. . . . . . . . . . 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 2760	. . . . . . . . 9 ⊢ (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
19	18	fveq2i 6838	. . . . . . . 8 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
20	13, 19	eqtr4di 2790	. . . . . . 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 6839	. . . . . 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 2772	. . . . 5 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
23	22	unieqd 4877	. . . 4 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24		fibas 22925	. . . . 5 ⊢ (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25		unitg 22915	. . . . 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 2788	. . 3 ⊢ (𝜑 → ∪ (∏t‘𝐹) = ∪ (fi‘(ran 𝑆 ∪ {𝑋})))
28		eqid 2737	. . . . . 6 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
29	28	ptuni 23542	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
30	1, 10, 29	syl2anc 585	. . . 4 ⊢ (𝜑 → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ (∏t‘𝐹))
31	11, 30	eqtrid 2784	. . 3 ⊢ (𝜑 → 𝑋 = ∪ (∏t‘𝐹))
32		ptcmp.5	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
33	32	pwexd 5325	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
34		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
35	34	mptpreima 6197	. . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38		elpw2g 5279	. . . . . . . . . . 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 8013	. . . . . . . 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 6671	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
45	33, 44	ssexd 5270	. . . . 5 ⊢ (𝜑 → ran 𝑆 ∈ V)
46		snex 5382	. . . . 5 ⊢ {𝑋} ∈ V
47		unexg 7690	. . . . 5 ⊢ ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
48	45, 46, 47	sylancl 587	. . . 4 ⊢ (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49		fiuni 9335	. . . 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 2782	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4555  {csn 4581  ∪ cuni 4864  ∪ ciun 4947   ↦ cmpt 5180   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493   ∈ cmpo 7362  Xcixp 8839  Fincfn 8887  ficfi 9317  cardccrd 9851  topGenctg 17361  ∏_tcpt 17362  Topctop 22841  TopBasesctb 22893  Compccmp 23334  UFLcufl 23848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8399  df-2o 8400  df-ixp 8840  df-en 8888  df-dom 8889  df-fin 8891  df-fi 9318  df-topgen 17367  df-pt 17368  df-top 22842  df-bases 22894  df-cmp 23335

This theorem is referenced by:  ptcmplem5  24004