Theorem ptcmplem5 24085

Description: Lemma for ptcmp 24087. (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 4227	. 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 24081	. . 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 4629	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ran 𝑆 → 𝑦 ⊆ ran 𝑆)
11	5	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝐴 ∈ 𝑉)
12	6	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝐹:𝐴⟶Comp)
13	1	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝑋 ∈ (UFL ∩ dom card))
14		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝑦 ⊆ ran 𝑆)
15		simplrr 777	. . . . . . . . 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 6085	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
18	17	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) ∈ 𝑦 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑦))
19	18	cbvrabv 3454	. . . . . . . . 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 24084	. . . . . . . 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 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
25	24	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
26		velpw 4627	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
27		eldif 3986	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
28		elpwunsn 4707	. . . . . . . 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 581	. . . . . 6 ⊢ ((𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
31	30	adantll 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
32		snssi 4833	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → {𝑋} ⊆ 𝑦)
33	32	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ⊆ 𝑦)
34		snfi 9109	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
35		elfpw 9424	. . . . . . . 8 ⊢ ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} ⊆ 𝑦 ∧ {𝑋} ∈ Fin))
36	33, 34, 35	sylanblrc 589	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
37		unisng 4949	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → ∪ {𝑋} = 𝑋)
38	37	eqcomd 2746	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑦 → 𝑋 = ∪ {𝑋})
39	38	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → 𝑋 = ∪ {𝑋})
40		unieq 4942	. . . . . . . 8 ⊢ (𝑧 = {𝑋} → ∪ 𝑧 = ∪ {𝑋})
41	40	rspceeqv 3658	. . . . . . 7 ⊢ (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = ∪ {𝑋}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
42	36, 39, 41	syl2anc 583	. . . . . 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 590	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
45	25, 44	pm2.61dan 812	. . 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 24074	1 ⊢ (𝜑 → (∏t‘𝐹) ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703  ⟶wf 6569  ‘cfv 6573   ∈ cmpo 7450  Xcixp 8955  Fincfn 9003  ficfi 9479  cardccrd 10004  topGenctg 17497  ∏_tcpt 17498  Compccmp 23415  UFLcufl 23929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-fin 9007  df-fi 9480  df-wdom 9634  df-card 10008  df-acn 10011  df-topgen 17503  df-pt 17504  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-cmp 23416  df-fil 23875  df-ufil 23930  df-ufl 23931  df-flim 23968  df-fcls 23970

This theorem is referenced by:  ptcmpg  24086