Theorem fin23lem39 10377

Description: Lemma for fin23 10416. Thus, we have that 𝑔 could not have been in 𝐹 after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.f	⊢ (𝜑 → ℎ:ω–1-1→V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
fin23lem.i	⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)

Assertion

Ref	Expression
fin23lem39	⊢ (𝜑 → ¬ 𝐺 ∈ 𝐹)

Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥,ℎ,𝐺   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗

Allowed substitution hints:   𝜑(𝑥, 𝑔, ℎ, 𝑖)   𝐹(𝑥, 𝑔, ℎ, 𝑖, 𝑗)   𝑌(𝑥, 𝑔, ℎ, 𝑖)

Proof of Theorem fin23lem39

Dummy variables 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem33.f	. . 3 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
2		fin23lem.f	. . 3 ⊢ (𝜑 → ℎ:ω–1-1→V)
3		fin23lem.g	. . 3 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
4		fin23lem.h	. . 3 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
5		fin23lem.i	. . 3 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
6	1, 2, 3, 4, 5	fin23lem38 10376	. 2 ⊢ (𝜑 → ¬ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
7	1, 2, 3, 4, 5	fin23lem35 10374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊊ ∪ ran (𝑌‘𝑒))
8	7	pssssd 4048	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒))
9		peano2 7892	. . . . . . . . 9 ⊢ (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑐 = suc 𝑒 → (𝑌‘𝑐) = (𝑌‘suc 𝑒))
11	10	rneqd 5924	. . . . . . . . . . 11 ⊢ (𝑐 = suc 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘suc 𝑒))
12	11	unieqd 4880	. . . . . . . . . 10 ⊢ (𝑐 = suc 𝑒 → ∪ ran (𝑌‘𝑐) = ∪ ran (𝑌‘suc 𝑒))
13		eqid 2760	. . . . . . . . . 10 ⊢ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))
14		fvex 6894	. . . . . . . . . . . 12 ⊢ (𝑌‘suc 𝑒) ∈ V
15	14	rnex 7913	. . . . . . . . . . 11 ⊢ ran (𝑌‘suc 𝑒) ∈ V
16	15	uniex 7749	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑒) ∈ V
17	12, 13, 16	fvmpt 6989	. . . . . . . . 9 ⊢ (suc 𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) = ∪ ran (𝑌‘suc 𝑒))
18	9, 17	syl 18	. . . . . . . 8 ⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) = ∪ ran (𝑌‘suc 𝑒))
19		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑒 → (𝑌‘𝑐) = (𝑌‘𝑒))
20	19	rneqd 5924	. . . . . . . . . 10 ⊢ (𝑐 = 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘𝑒))
21	20	unieqd 4880	. . . . . . . . 9 ⊢ (𝑐 = 𝑒 → ∪ ran (𝑌‘𝑐) = ∪ ran (𝑌‘𝑒))
22		fvex 6894	. . . . . . . . . . 11 ⊢ (𝑌‘𝑒) ∈ V
23	22	rnex 7913	. . . . . . . . . 10 ⊢ ran (𝑌‘𝑒) ∈ V
24	23	uniex 7749	. . . . . . . . 9 ⊢ ∪ ran (𝑌‘𝑒) ∈ V
25	21, 13, 24	fvmpt 6989	. . . . . . . 8 ⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) = ∪ ran (𝑌‘𝑒))
26	18, 25	sseq12d 3964	. . . . . . 7 ⊢ (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) ↔ ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒)))
27	26	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) ↔ ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒)))
28	8, 27	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))
29	28	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))
30	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))
31		fveq1 6880	. . . . . . 7 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒))
32		fveq1 6880	. . . . . . 7 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))
33	31, 32	sseq12d 3964	. . . . . 6 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) ↔ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)))
34	33	ralbidv 3185	. . . . 5 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) ↔ ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒)))
35		rneq 5922	. . . . . . 7 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
36	35	inteqd 4912	. . . . . 6 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ∩ ran 𝑑 = ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
37	36, 35	eleq12d 2854	. . . . 5 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (∩ ran 𝑑 ∈ ran 𝑑 ↔ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))))
38	34, 37	imbi12d 347	. . . 4 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ((∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran 𝑑 ∈ ran 𝑑) ↔ (∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) → ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))))
39	1	isfin3ds 10356	. . . . . 6 ⊢ (𝐺 ∈ 𝐹 → (𝐺 ∈ 𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran 𝑑 ∈ ran 𝑑)))
40	39	ibi 270	. . . . 5 ⊢ (𝐺 ∈ 𝐹 → ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran 𝑑 ∈ ran 𝑑))
41	40	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∀𝑑 ∈ (𝒫 𝐺 ↑m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑‘𝑒) → ∩ ran 𝑑 ∈ ran 𝑑))
42	1, 2, 3, 4, 5	fin23lem34 10373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ((𝑌‘𝑐):ω–1-1→V ∧ ∪ ran (𝑌‘𝑐) ⊆ 𝐺))
43	42	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺)
44	43	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ⊆ 𝐺)
45		elpw2g 5298	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐹 → (∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺 ↔ ∪ ran (𝑌‘𝑐) ⊆ 𝐺))
46	45	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → (∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺 ↔ ∪ ran (𝑌‘𝑐) ⊆ 𝐺))
47	44, 46	mpbird 260	. . . . . 6 ⊢ (((𝜑 ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ ω) → ∪ ran (𝑌‘𝑐) ∈ 𝒫 𝐺)
48	47	fmpttd 7111	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺)
49		pwexg 5343	. . . . . 6 ⊢ (𝐺 ∈ 𝐹 → 𝒫 𝐺 ∈ V)
50		vex 3454	. . . . . . . 8 ⊢ ℎ ∈ V
51		f1f 6774	. . . . . . . 8 ⊢ (ℎ:ω–1-1→V → ℎ:ω⟶V)
52		dmfex 7908	. . . . . . . 8 ⊢ ((ℎ ∈ V ∧ ℎ:ω⟶V) → ω ∈ V)
53	50, 51, 52	sylancr 599	. . . . . . 7 ⊢ (ℎ:ω–1-1→V → ω ∈ V)
54	2, 53	syl 18	. . . . . 6 ⊢ (𝜑 → ω ∈ V)
55		elmapg 8845	. . . . . 6 ⊢ ((𝒫 𝐺 ∈ V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m ω) ↔ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺))
56	49, 54, 55	syl2anr 609	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m ω) ↔ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)):ω⟶𝒫 𝐺))
57	48, 56	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ (𝒫 𝐺 ↑m ω))
58	38, 41, 57	rspcdva 3577	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → (∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) → ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))))
59	30, 58	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐹) → ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
60	6, 59	mtand 828	1 ⊢ (𝜑 → ¬ 𝐺 ∈ 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3899   ⊊ wpss 3900  𝒫 cpw 4557  ∪ cuni 4867  ∩ cint 4907   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657  suc csuc 6361  ⟶wf 6531  –1-1→wf1 6532  ‘cfv 6535  (class class class)co 7416  ωcom 7868  reccrdg 8403   ↑_m cmap 8833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-map 8835

This theorem is used by:  fin23lem41  10379