Theorem fin23lem39 10355

Description: Lemma for fin23 10394. 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 10354	. 2 ⊢ (𝜑 → ¬ ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) ∈ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
7	1, 2, 3, 4, 5	fin23lem35 10352	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊊ ∪ ran (𝑌‘𝑒))
8	7	pssssd 4051	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ω) → ∪ ran (𝑌‘suc 𝑒) ⊆ ∪ ran (𝑌‘𝑒))
9		peano2 7889	. . . . . . . . 9 ⊢ (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑐 = suc 𝑒 → (𝑌‘𝑐) = (𝑌‘suc 𝑒))
11	10	rneqd 5926	. . . . . . . . . . 11 ⊢ (𝑐 = suc 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘suc 𝑒))
12	11	unieqd 4883	. . . . . . . . . 10 ⊢ (𝑐 = suc 𝑒 → ∪ ran (𝑌‘𝑐) = ∪ ran (𝑌‘suc 𝑒))
13		eqid 2762	. . . . . . . . . 10 ⊢ (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))
14		fvex 6895	. . . . . . . . . . . 12 ⊢ (𝑌‘suc 𝑒) ∈ V
15	14	rnex 7910	. . . . . . . . . . 11 ⊢ ran (𝑌‘suc 𝑒) ∈ V
16	15	uniex 7746	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑒) ∈ V
17	12, 13, 16	fvmpt 6990	. . . . . . . . 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 6882	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑒 → (𝑌‘𝑐) = (𝑌‘𝑒))
20	19	rneqd 5926	. . . . . . . . . 10 ⊢ (𝑐 = 𝑒 → ran (𝑌‘𝑐) = ran (𝑌‘𝑒))
21	20	unieqd 4883	. . . . . . . . 9 ⊢ (𝑐 = 𝑒 → ∪ ran (𝑌‘𝑐) = ∪ ran (𝑌‘𝑒))
22		fvex 6895	. . . . . . . . . . 11 ⊢ (𝑌‘𝑒) ∈ V
23	22	rnex 7910	. . . . . . . . . 10 ⊢ ran (𝑌‘𝑒) ∈ V
24	23	uniex 7746	. . . . . . . . 9 ⊢ ∪ ran (𝑌‘𝑒) ∈ V
25	21, 13, 24	fvmpt 6990	. . . . . . . 8 ⊢ (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒) = ∪ ran (𝑌‘𝑒))
26	18, 25	sseq12d 3967	. . . . . . 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 3156	. . . 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 6881	. . . . . . 7 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘suc 𝑒))
32		fveq1 6881	. . . . . . 7 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → (𝑑‘𝑒) = ((𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐))‘𝑒))
33	31, 32	sseq12d 3967	. . . . . 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 3187	. . . . 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 5924	. . . . . . 7 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
36	35	inteqd 4915	. . . . . 6 ⊢ (𝑑 = (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)) → ∩ ran 𝑑 = ∩ ran (𝑐 ∈ ω ↦ ∪ ran (𝑌‘𝑐)))
37	36, 35	eleq12d 2856	. . . . 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 10334	. . . . . 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 10351	. . . . . . . . 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 5302	. . . . . . . 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 5347	. . . . . 6 ⊢ (𝐺 ∈ 𝐹 → 𝒫 𝐺 ∈ V)
50		vex 3457	. . . . . . . 8 ⊢ ℎ ∈ V
51		f1f 6775	. . . . . . . 8 ⊢ (ℎ:ω–1-1→V → ℎ:ω⟶V)
52		dmfex 7905	. . . . . . . 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 8841	. . . . . 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 3580	. . 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 2740  ∀wral 3078  Vcvv 3453   ⊆ wss 3902   ⊊ wpss 3903  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661  suc csuc 6363  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7416  ωcom 7865  reccrdg 8401   ↑_m cmap 8829

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-map 8831

This theorem is used by:  fin23lem41  10357