Theorem cnfcom3clem 9601

Description: Lemma for cnfcom3c 9602. (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 1136	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3		omelon 9542	. . . . . . . . 9 ⊢ ω ∈ On
4		1onn 8558	. . . . . . . . 9 ⊢ 1o ∈ ω
5		ondif2 8420	. . . . . . . . 9 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
6	3, 4, 5	mpbir2an 711	. . . . . . . 8 ⊢ ω ∈ (On ∖ 2o)
7		oeworde 8511	. . . . . . . 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 1137	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑏 ∈ 𝐴)
10	8, 9	sseldd 3936	. . . . . 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 1138	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
19	1, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18	cnfcom3lem 9599	. . . . 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 9600	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏–1-1-onto→(ω ↑o 𝑊))
24		f1of 6764	. . . . . . . . . 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 7163	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
27		cnfcom3c.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
28	27	fvmpt2 6941	. . . . . . . 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 6759	. . . . . 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 257	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏) → (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊))
32		oveq2 7357	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (ω ↑o 𝑤) = (ω ↑o 𝑊))
33	32	f1oeq3d 6761	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
34	33	rspcev 3577	. . . . 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 1121	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
37	36	ralrimiva 3121	. 2 ⊢ (𝐴 ∈ On → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
38		ovex 7382	. . . . 5 ⊢ (ω ↑o 𝐴) ∈ V
39	38	mptex 7159	. . . 4 ⊢ (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ∈ V
40	27, 39	eqeltri 2824	. . 3 ⊢ 𝐿 ∈ V
41		nfmpt1 5191	. . . . . 6 ⊢ Ⅎ𝑏(𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
42	27, 41	nfcxfr 2889	. . . . 5 ⊢ Ⅎ𝑏𝐿
43	42	nfeq2 2909	. . . 4 ⊢ Ⅎ𝑏 𝑔 = 𝐿
44		fveq1 6821	. . . . . . 7 ⊢ (𝑔 = 𝐿 → (𝑔‘𝑏) = (𝐿‘𝑏))
45	44	f1oeq1d 6759	. . . . . 6 ⊢ (𝑔 = 𝐿 → ((𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) ↔ (𝐿‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
46	45	rexbidv 3153	. . . . 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 340	. . . 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 3242	. . 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 3561	. 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 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  ∪ cuni 4858   ↦ cmpt 5173   E cep 5518  ^◡ccnv 5618  dom cdm 5619   ∘ ccom 5623  Oncon0 6307  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  ωcom 7799   supp csupp 8093  seq_ωcseqom 8369  1_oc1o 8381  2_oc2o 8382   +_o coa 8385   ·_o comu 8386   ↑_o coe 8387  OrdIsocoi 9401   CNF ccnf 9557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-seqom 8370  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-oexp 8394  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-oi 9402  df-cnf 9558

This theorem is referenced by:  cnfcom3c  9602