Theorem infxpenc2lem2 10004

Description: Lemma for infxpenc2 10006. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)

Hypotheses

Ref	Expression
infxpenc2.1	⊢ (𝜑 → 𝐴 ∈ On)
infxpenc2.2	⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
infxpenc2.3	⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))
infxpenc2.4	⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
infxpenc2.5	⊢ (𝜑 → (𝐹‘∅) = ∅)
infxpenc2.k	⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
infxpenc2.h	⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))
infxpenc2.l	⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
infxpenc2.x	⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
infxpenc2.y	⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
infxpenc2.j	⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))
infxpenc2.z	⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
infxpenc2.t	⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
infxpenc2.g	⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))

Assertion

Ref	Expression
infxpenc2lem2	⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Distinct variable groups:   𝑔,𝑏,𝑛,𝑤,𝑥,𝑦,𝐴   𝜑,𝑏,𝑤,𝑥,𝑦   𝑧,𝑔,𝑊,𝑤,𝑥,𝑦   𝑔,𝐹,𝑥,𝑦   𝑔,𝐺   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑔,𝑛)   𝐴(𝑧)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐹(𝑧,𝑤,𝑛,𝑏)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑛,𝑏)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝑊(𝑛,𝑏)   𝑋(𝑧,𝑤,𝑔,𝑛,𝑏)   𝑌(𝑧,𝑤,𝑔,𝑛,𝑏)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)

Proof of Theorem infxpenc2lem2

Step	Hyp	Ref	Expression
1		infxpenc2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
2	1	mptexd 7223	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐴 ↦ 𝐺) ∈ V)
3	1	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐴 ∈ On)
4		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ 𝐴)
5		onelon 6386	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ On)
6	3, 4, 5	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ On)
7		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ω ⊆ 𝑏)
8		infxpenc2.2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
9		infxpenc2.3	. . . . . . . 8 ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))
10	1, 8, 9	infxpenc2lem1 10003	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
11	10	simpld 499	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝑊 ∈ (On ∖ 1o))
12		infxpenc2.4	. . . . . . 7 ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
13	12	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
14		infxpenc2.5	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
15	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝐹‘∅) = ∅)
16	10	simprd 500	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊))
17		infxpenc2.k	. . . . . 6 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
18		infxpenc2.h	. . . . . 6 ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))
19		infxpenc2.l	. . . . . 6 ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
20		infxpenc2.x	. . . . . 6 ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
21		infxpenc2.y	. . . . . 6 ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
22		infxpenc2.j	. . . . . 6 ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))
23		infxpenc2.z	. . . . . 6 ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
24		infxpenc2.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)
25		infxpenc2.g	. . . . . 6 ⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
26	6, 7, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	infxpenc 10002	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏)
27		f1of 6821	. . . . . . . . 9 ⊢ (𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏 → 𝐺:(𝑏 × 𝑏)⟶𝑏)
28	26, 27	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)⟶𝑏)
29		vex 3467	. . . . . . . . 9 ⊢ 𝑏 ∈ V
30	29, 29	xpex 7752	. . . . . . . 8 ⊢ (𝑏 × 𝑏) ∈ V
31		fex 7225	. . . . . . . 8 ⊢ ((𝐺:(𝑏 × 𝑏)⟶𝑏 ∧ (𝑏 × 𝑏) ∈ V) → 𝐺 ∈ V)
32	28, 30, 31	sylancl 597	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → 𝐺 ∈ V)
33		eqid 2769	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐴 ↦ 𝐺) = (𝑏 ∈ 𝐴 ↦ 𝐺)
34	33	fvmpt2 7002	. . . . . . 7 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝐺 ∈ V) → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏) = 𝐺)
35	4, 32, 34	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏) = 𝐺)
36	35	f1oeq1d 6816	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ 𝐺:(𝑏 × 𝑏)–1-1-onto→𝑏))
37	26, 36	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)
38	37	expr 461	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
39	38	ralrimiva 3163	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
40		nfmpt1 5214	. . . 4 ⊢ Ⅎ𝑏(𝑏 ∈ 𝐴 ↦ 𝐺)
41	40	nfeq2 2948	. . 3 ⊢ Ⅎ𝑏 𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺)
42		fveq1 6881	. . . . 5 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → (𝑔‘𝑏) = ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏))
43	42	f1oeq1d 6816	. . . 4 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → ((𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
44	43	imbi2d 343	. . 3 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → ((ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
45	41, 44	ralbid 3284	. 2 ⊢ (𝑔 = (𝑏 ∈ 𝐴 ↦ 𝐺) → (∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ((𝑏 ∈ 𝐴 ↦ 𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
46	2, 39, 45	spcedv 3566	1 ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ⟨cop 4600   class class class wbr 5113   ↦ cmpt 5196   I cid 5556   × cxp 5660  ^◡ccnv 5661  ran crn 5663   ↾ cres 5664   ∘ ccom 5666  Oncon0 6361  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  ωcom 7862  1_oc1o 8446  2_oc2o 8447   +_o coa 8450   ·_o comu 8451   ↑_o coe 8452   ↑_m cmap 8824   finSupp cfsupp 9321   CNF ccnf 9630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-seqom 8435  df-1o 8453  df-2o 8454  df-oadd 8457  df-omul 8458  df-oexp 8459  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-oi 9472  df-cnf 9631

This theorem is referenced by:  infxpenc2lem3  10005