Theorem ptcmplem5 24079

Description: Lemma for ptcmp 24081. (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
ptcmplem5	⊢ (𝜑 → (∏t‘𝐹) ∈ Comp)

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 ptcmplem5

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

Step	Hyp	Ref	Expression
1		ptcmp.5	. . 3 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
2	1	elin1d 4213	. 2 ⊢ (𝜑 → 𝑋 ∈ UFL)
3		ptcmp.1	. . . 4 ⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
4		ptcmp.2	. . . 4 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
5		ptcmp.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		ptcmp.4	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶Comp)
7	3, 4, 5, 6, 1	ptcmplem1 24075	. . 3 ⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
8	7	simpld 494	. 2 ⊢ (𝜑 → 𝑋 = ∪ (ran 𝑆 ∪ {𝑋}))
9	7	simprd 495	. 2 ⊢ (𝜑 → (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
10		elpwi 4611	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ran 𝑆 → 𝑦 ⊆ ran 𝑆)
11	5	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝐴 ∈ 𝑉)
12	6	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝐹:𝐴⟶Comp)
13	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝑋 ∈ (UFL ∩ dom card))
14		simplrl 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝑦 ⊆ ran 𝑆)
15		simplrr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝑋 = ∪ 𝑦)
16		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
17		imaeq2 6075	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
18	17	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) ∈ 𝑦 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑦))
19	18	cbvrabv 3443	. . . . . . . . 9 ⊢ {𝑧 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) ∈ 𝑦} = {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑦}
20	3, 4, 11, 12, 13, 14, 15, 16, 19	ptcmplem4 24078	. . . . . . . 8 ⊢ ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
21		iman 401	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) ↔ ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
22	20, 21	mpbir 231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
23	22	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ⊆ ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
24	10, 23	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
25	24	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
26		velpw 4609	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
27		eldif 3972	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
28		elpwunsn 4688	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
29	27, 28	sylbir 235	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
30	26, 29	sylanbr 582	. . . . . 6 ⊢ ((𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
31	30	adantll 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
32		snssi 4812	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → {𝑋} ⊆ 𝑦)
33	32	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ⊆ 𝑦)
34		snfi 9081	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
35		elfpw 9391	. . . . . . . 8 ⊢ ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} ⊆ 𝑦 ∧ {𝑋} ∈ Fin))
36	33, 34, 35	sylanblrc 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
37		unisng 4929	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → ∪ {𝑋} = 𝑋)
38	37	eqcomd 2740	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑦 → 𝑋 = ∪ {𝑋})
39	38	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → 𝑋 = ∪ {𝑋})
40		unieq 4922	. . . . . . . 8 ⊢ (𝑧 = {𝑋} → ∪ 𝑧 = ∪ {𝑋})
41	40	rspceeqv 3644	. . . . . . 7 ⊢ (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = ∪ {𝑋}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
42	36, 39, 41	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
43	42	a1d 25	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
44	31, 43	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
45	25, 44	pm2.61dan 813	. . 3 ⊢ ((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
46	45	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ 𝑋 = ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
47	2, 8, 9, 46	alexsub 24068	1 ⊢ (𝜑 → (∏t‘𝐹) ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  {crab 3432   ∖ cdif 3959   ∪ cun 3960   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  {csn 4630  ∪ cuni 4911   ↦ cmpt 5230  ^◡ccnv 5687  dom cdm 5688  ran crn 5689   “ cima 5691  ⟶wf 6558  ‘cfv 6562   ∈ cmpo 7432  Xcixp 8935  Fincfn 8983  ficfi 9447  cardccrd 9972  topGenctg 17483  ∏_tcpt 17484  Compccmp 23409  UFLcufl 23923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-fin 8987  df-fi 9448  df-wdom 9602  df-card 9976  df-acn 9979  df-topgen 17489  df-pt 17490  df-fbas 21378  df-fg 21379  df-top 22915  df-topon 22932  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-cmp 23410  df-fil 23869  df-ufil 23924  df-ufl 23925  df-flim 23962  df-fcls 23964

This theorem is referenced by:  ptcmpg  24080