Theorem ptcmplem5 24017

Description: Lemma for ptcmp 24019. (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 4158	. 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 24013	. . 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 4563	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ran 𝑆 → 𝑦 ⊆ ran 𝑆)
11	5	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝐴 ∈ 𝑉)
12	6	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ ran 𝑆 ∧ 𝑋 = ∪ 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → 𝐹:𝐴⟶Comp)
13	1	ad2antrr 727	. . . . . . . . 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 6025	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
18	17	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑧) ∈ 𝑦 ↔ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑦))
19	18	cbvrabv 3411	. . . . . . . . 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 24016	. . . . . . . 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 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
25	24	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
26		velpw 4561	. . . . . . 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 4643	. . . . . . . 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 583	. . . . . 6 ⊢ ((𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
31	30	adantll 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋 ∈ 𝑦)
32		snssi 4766	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → {𝑋} ⊆ 𝑦)
33	32	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ⊆ 𝑦)
34		snfi 8994	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
35		elfpw 9268	. . . . . . . 8 ⊢ ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} ⊆ 𝑦 ∧ {𝑋} ∈ Fin))
36	33, 34, 35	sylanblrc 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
37		unisng 4883	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑦 → ∪ {𝑋} = 𝑋)
38	37	eqcomd 2743	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑦 → 𝑋 = ∪ {𝑋})
39	38	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋 ∈ 𝑦) → 𝑋 = ∪ {𝑋})
40		unieq 4876	. . . . . . . 8 ⊢ (𝑧 = {𝑋} → ∪ 𝑧 = ∪ {𝑋})
41	40	rspceeqv 3601	. . . . . . 7 ⊢ (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = ∪ {𝑋}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)
42	36, 39, 41	syl2anc 585	. . . . . 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 592	. . . 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 24006	1 ⊢ (𝜑 → (∏t‘𝐹) ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181  ^◡ccnv 5633  dom cdm 5634  ran crn 5635   “ cima 5637  ⟶wf 6498  ‘cfv 6502   ∈ cmpo 7372  Xcixp 8849  Fincfn 8897  ficfi 9327  cardccrd 9861  topGenctg 17371  ∏_tcpt 17372  Compccmp 23347  UFLcufl 23861

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-omul 8414  df-er 8647  df-map 8779  df-ixp 8850  df-en 8898  df-dom 8899  df-fin 8901  df-fi 9328  df-wdom 9484  df-card 9865  df-acn 9868  df-topgen 17377  df-pt 17378  df-fbas 21323  df-fg 21324  df-top 22855  df-topon 22872  df-bases 22907  df-cld 22980  df-ntr 22981  df-cls 22982  df-nei 23059  df-cmp 23348  df-fil 23807  df-ufil 23862  df-ufl 23863  df-flim 23900  df-fcls 23902

This theorem is referenced by:  ptcmpg  24018