Theorem ptcmplem5 23941

Description: Lemma for ptcmp 23943. (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 4155	. 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 23937	. . 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 4558	. . . . . 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 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 6007	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
18	17	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) ∈ 𝑦 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑦))
19	18	cbvrabv 3405	. . . . . . . . 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 23940	. . . . . . . 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 4556	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
27		eldif 3913	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
28		elpwunsn 4636	. . . . . . . 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 4759	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → {𝑋} ⊆ 𝑦)
33	32	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ⊆ 𝑦)
34		snfi 8968	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
35		elfpw 9244	. . . . . . . 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 4876	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → ∪ {𝑋} = 𝑋)
38	37	eqcomd 2735	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑦 → 𝑋 = ∪ {𝑋})
39	38	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → 𝑋 = ∪ {𝑋})
40		unieq 4869	. . . . . . . 8 ⊢ (𝑧 = {𝑋} → ∪ 𝑧 = ∪ {𝑋})
41	40	rspceeqv 3600	. . . . . . 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 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 23930	1 ⊢ (𝜑 → (∏t‘𝐹) ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3394   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  {csn 4577  ∪ cuni 4858   ↦ cmpt 5173  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  ⟶wf 6478  ‘cfv 6482   ∈ cmpo 7351  Xcixp 8824  Fincfn 8872  ficfi 9300  cardccrd 9831  topGenctg 17341  ∏_tcpt 17342  Compccmp 23271  UFLcufl 23785

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-fin 8876  df-fi 9301  df-wdom 9457  df-card 9835  df-acn 9838  df-topgen 17347  df-pt 17348  df-fbas 21258  df-fg 21259  df-top 22779  df-topon 22796  df-bases 22831  df-cld 22904  df-ntr 22905  df-cls 22906  df-nei 22983  df-cmp 23272  df-fil 23731  df-ufil 23786  df-ufl 23787  df-flim 23824  df-fcls 23826

This theorem is referenced by:  ptcmpg  23942