Theorem cnfcom3clem 9463

Description: Lemma for cnfcom3c 9464. (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 1135	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3		omelon 9404	. . . . . . . . 9 ⊢ ω ∈ On
4		1onn 8470	. . . . . . . . 9 ⊢ 1o ∈ ω
5		ondif2 8332	. . . . . . . . 9 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
6	3, 4, 5	mpbir2an 708	. . . . . . . 8 ⊢ ω ∈ (On ∖ 2o)
7		oeworde 8424	. . . . . . . 8 ⊢ ((ω ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ⊆ (ω ↑o 𝐴))
8	6, 2, 7	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ⊆ (ω ↑o 𝐴))
9		simp2 1136	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑏 ∈ 𝐴)
10	8, 9	sseldd 3922	. . . . . 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 1137	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
19	1, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18	cnfcom3lem 9461	. . . . 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 9462	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏–1-1-onto→(ω ↑o 𝑊))
24		f1of 6716	. . . . . . . . . 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	25, 9	fexd 7103	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
27		cnfcom3c.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
28	27	fvmpt2 6886	. . . . . . . 8 ⊢ ((𝑏 ∈ (ω ↑o 𝐴) ∧ 𝑁 ∈ V) → (𝐿‘𝑏) = 𝑁)
29	10, 26, 28	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → (𝐿‘𝑏) = 𝑁)
30	29	f1oeq1d 6711	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊) ↔ 𝑁:𝑏–1-1-onto→(ω ↑o 𝑊)))
31	23, 30	mpbird 256	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊))
32		oveq2 7283	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (ω ↑o 𝑤) = (ω ↑o 𝑊))
33	32	f1oeq3d 6713	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
34	33	rspcev 3561	. . . . 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))
35	19, 31, 34	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ∃𝑤 ∈ (On ∖ 1o)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))
36	35	3expia 1120	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
37	36	ralrimiva 3103	. 2 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
38		ovex 7308	. . . . 5 ⊢ (ω ↑o 𝐴) ∈ V
39	38	mptex 7099	. . . 4 ⊢ (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ∈ V
40	27, 39	eqeltri 2835	. . 3 ⊢ 𝐿 ∈ V
41		nfmpt1 5182	. . . . . 6 ⊢ Ⅎ𝑏(𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
42	27, 41	nfcxfr 2905	. . . . 5 ⊢ Ⅎ𝑏𝐿
43	42	nfeq2 2924	. . . 4 ⊢ Ⅎ𝑏 𝑔 = 𝐿
44		fveq1 6773	. . . . . . 7 ⊢ (𝑔 = 𝐿 → (𝑔‘𝑏) = (𝐿‘𝑏))
45	44	f1oeq1d 6711	. . . . . 6 ⊢ (𝑔 = 𝐿 → ((𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
46	45	rexbidv 3226	. . . . 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)))
47	46	imbi2d 341	. . . 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))))
48	43, 47	ralbid 3161	. . 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))))
49	40, 48	spcev 3545	. 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)))
50	37, 49	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   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839   ↦ cmpt 5157   E cep 5494  ^◡ccnv 5588  dom cdm 5589   ∘ ccom 5593  Oncon0 6266  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712   supp csupp 7977  seq_ωcseqom 8278  1_oc1o 8290  2_oc2o 8291   +_o coa 8294   ·_o comu 8295   ↑_o coe 8296  OrdIsocoi 9268   CNF ccnf 9419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-2o 8298  df-oadd 8301  df-omul 8302  df-oexp 8303  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-cnf 9420

This theorem is referenced by:  cnfcom3c  9464