Theorem cnfcom3clem 8485

Description: Lemma for cnfcom3c 8486. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)

Hypotheses

Ref	Expression
cnfcom3c.s	⊢ 𝑆 = dom (ω CNF 𝐴)
cnfcom3c.f	⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝑏)
cnfcom3c.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom3c.h	⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
cnfcom3c.t	⊢ 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom3c.m	⊢ 𝑀 = ((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘)))
cnfcom3c.k	⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
cnfcom3c.w	⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
cnfcom3c.x	⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹‘𝑊) ·𝑜 𝑣) +𝑜 𝑢))
cnfcom3c.y	⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))
cnfcom3c.n	⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))
cnfcom3c.l	⊢ 𝐿 = (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)

Assertion

Ref	Expression
cnfcom3clem	⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))

Distinct variable groups:   𝑔,𝑏,𝑘,𝑢,𝑣,𝑤,𝑥,𝑧,𝐴   𝑢,𝐾,𝑣   𝑔,𝐿,𝑤   𝑥,𝑀   𝑢,𝑇,𝑣,𝑧   𝑓,𝑘,𝑢,𝑣,𝑥,𝑧,𝐹   𝑓,𝐺,𝑘,𝑢,𝑣,𝑥,𝑧   𝑓,𝐻,𝑢,𝑣,𝑥   𝑆,𝑘,𝑧   𝑢,𝑊,𝑣,𝑤,𝑥

Allowed substitution hints:   𝐴(𝑓)   𝑆(𝑥,𝑤,𝑣,𝑢,𝑓,𝑔,𝑏)   𝑇(𝑥,𝑤,𝑓,𝑔,𝑘,𝑏)   𝐹(𝑤,𝑔,𝑏)   𝐺(𝑤,𝑔,𝑏)   𝐻(𝑧,𝑤,𝑔,𝑘,𝑏)   𝐾(𝑥,𝑧,𝑤,𝑓,𝑔,𝑘,𝑏)   𝐿(𝑥,𝑧,𝑣,𝑢,𝑓,𝑘,𝑏)   𝑀(𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)   𝑁(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)   𝑊(𝑧,𝑓,𝑔,𝑘,𝑏)   𝑋(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)   𝑌(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)

Proof of Theorem cnfcom3clem

Step	Hyp	Ref	Expression
1		cnfcom3c.s	. . . . . 6 ⊢ 𝑆 = dom (ω CNF 𝐴)
2		simp1 1054	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3		omelon 8426	. . . . . . . . 9 ⊢ ω ∈ On
4		1onn 7606	. . . . . . . . 9 ⊢ 1𝑜 ∈ ω
5		ondif2 7469	. . . . . . . . 9 ⊢ (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
6	3, 4, 5	mpbir2an 957	. . . . . . . 8 ⊢ ω ∈ (On ∖ 2𝑜)
7		oeworde 7560	. . . . . . . 8 ⊢ ((ω ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ⊆ (ω ↑𝑜 𝐴))
8	6, 2, 7	sylancr 694	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ⊆ (ω ↑𝑜 𝐴))
9		simp2 1055	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑏 ∈ 𝐴)
10	8, 9	sseldd 3569	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑏 ∈ (ω ↑𝑜 𝐴))
11		cnfcom3c.f	. . . . . 6 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝑏)
12		cnfcom3c.g	. . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
13		cnfcom3c.h	. . . . . 6 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
14		cnfcom3c.t	. . . . . 6 ⊢ 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
15		cnfcom3c.m	. . . . . 6 ⊢ 𝑀 = ((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘)))
16		cnfcom3c.k	. . . . . 6 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
17		cnfcom3c.w	. . . . . 6 ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
18		simp3 1056	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
19	1, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18	cnfcom3lem 8483	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑊 ∈ (On ∖ 1𝑜))
20		cnfcom3c.x	. . . . . . 7 ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹‘𝑊) ·𝑜 𝑣) +𝑜 𝑢))
21		cnfcom3c.y	. . . . . . 7 ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))
22		cnfcom3c.n	. . . . . . 7 ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))
23	1, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22	cnfcom3 8484	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏–1-1-onto→(ω ↑𝑜 𝑊))
24		f1of 6050	. . . . . . . . . 10 ⊢ (𝑁:𝑏–1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:𝑏⟶(ω ↑𝑜 𝑊))
25	23, 24	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏⟶(ω ↑𝑜 𝑊))
26		vex 3176	. . . . . . . . 9 ⊢ 𝑏 ∈ V
27		fex 6394	. . . . . . . . 9 ⊢ ((𝑁:𝑏⟶(ω ↑𝑜 𝑊) ∧ 𝑏 ∈ V) → 𝑁 ∈ V)
28	25, 26, 27	sylancl 693	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
29		cnfcom3c.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)
30	29	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑏 ∈ (ω ↑𝑜 𝐴) ∧ 𝑁 ∈ V) → (𝐿‘𝑏) = 𝑁)
31	10, 28, 30	syl2anc 691	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → (𝐿‘𝑏) = 𝑁)
32		f1oeq1 6040	. . . . . . 7 ⊢ ((𝐿‘𝑏) = 𝑁 → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊) ↔ 𝑁:𝑏–1-1-onto→(ω ↑𝑜 𝑊)))
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊) ↔ 𝑁:𝑏–1-1-onto→(ω ↑𝑜 𝑊)))
34	23, 33	mpbird 246	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊))
35		oveq2 6557	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (ω ↑𝑜 𝑤) = (ω ↑𝑜 𝑊))
36		f1oeq3 6042	. . . . . . 7 ⊢ ((ω ↑𝑜 𝑤) = (ω ↑𝑜 𝑊) → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)))
37	35, 36	syl 17	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)))
38	37	rspcev 3282	. . . . 5 ⊢ ((𝑊 ∈ (On ∖ 1𝑜) ∧ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))
39	19, 34, 38	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))
40	39	3expia 1259	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
41	40	ralrimiva 2949	. 2 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
42		ovex 6577	. . . . 5 ⊢ (ω ↑𝑜 𝐴) ∈ V
43	42	mptex 6390	. . . 4 ⊢ (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁) ∈ V
44	29, 43	eqeltri 2684	. . 3 ⊢ 𝐿 ∈ V
45		nfmpt1 4675	. . . . . 6 ⊢ Ⅎ𝑏(𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)
46	29, 45	nfcxfr 2749	. . . . 5 ⊢ Ⅎ𝑏𝐿
47	46	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑏 𝑔 = 𝐿
48		fveq1 6102	. . . . . . 7 ⊢ (𝑔 = 𝐿 → (𝑔‘𝑏) = (𝐿‘𝑏))
49		f1oeq1 6040	. . . . . . 7 ⊢ ((𝑔‘𝑏) = (𝐿‘𝑏) → ((𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
50	48, 49	syl 17	. . . . . 6 ⊢ (𝑔 = 𝐿 → ((𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
51	50	rexbidv 3034	. . . . 5 ⊢ (𝑔 = 𝐿 → (∃𝑤 ∈ (On ∖ 1𝑜)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
52	51	imbi2d 329	. . . 4 ⊢ (𝑔 = 𝐿 → ((ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)) ↔ (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))))
53	47, 52	ralbid 2966	. . 3 ⊢ (𝑔 = 𝐿 → (∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)) ↔ ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))))
54	44, 53	spcev 3273	. 2 ⊢ (∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
55	41, 54	syl 17	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   ↦ cmpt 4643   E cep 4947  ^◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq_𝜔cseqom 7429  1_𝑜c1o 7440  2_𝑜c2o 7441   +_𝑜 coa 7444   ·_𝑜 comu 7445   ↑_𝑜 coe 7446  OrdIsocoi 8297   CNF ccnf 8441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442

This theorem is referenced by:  cnfcom3c  8486