Theorem infxpenc2 9944

Description: Existence form of infxpenc 9940. A "uniform" or "canonical" version of infxpen 9936, asserting the existence of a single function 𝑔 that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)

Assertion

Ref	Expression
infxpenc2	⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Distinct variable group:   𝑔,𝑏,𝐴

Proof of Theorem infxpenc2

Dummy variables 𝑓 𝑛 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnfcom3c 9627	. 2 ⊢ (𝐴 ∈ On → ∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)))
2		df-2o 8408	. . . . . . . 8 ⊢ 2o = suc 1o
3	2	oveq2i 7379	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
4		omelon 9567	. . . . . . . 8 ⊢ ω ∈ On
5		1on 8419	. . . . . . . 8 ⊢ 1o ∈ On
6		oesuc 8464	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
7	4, 5, 6	mp2an 693	. . . . . . 7 ⊢ (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
8		oe1 8481	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
9	4, 8	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
10	9	oveq1i 7378	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
11	3, 7, 10	3eqtri 2764	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
12		omxpen 9019	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (ω ·o ω) ≈ (ω × ω))
13	4, 4, 12	mp2an 693	. . . . . 6 ⊢ (ω ·o ω) ≈ (ω × ω)
14	11, 13	eqbrtri 5121	. . . . 5 ⊢ (ω ↑o 2o) ≈ (ω × ω)
15		xpomen 9937	. . . . 5 ⊢ (ω × ω) ≈ ω
16	14, 15	entri 8957	. . . 4 ⊢ (ω ↑o 2o) ≈ ω
17	16	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (ω ↑o 2o) ≈ ω)
18		bren 8905	. . 3 ⊢ ((ω ↑o 2o) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
19	17, 18	sylib 218	. 2 ⊢ (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
20		exdistrv 1957	. . 3 ⊢ (∃𝑛∃𝑓(∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω) ↔ (∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω))
21		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → 𝐴 ∈ On)
22		simprl 771	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)))
23		sseq2 3962	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24		oveq2 7376	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (ω ↑o 𝑦) = (ω ↑o 𝑤))
25	24	f1oeq3d 6779	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ (𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤)))
26	25	cbvrexvw 3217	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤))
27		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝑛‘𝑥) = (𝑛‘𝑏))
28	27	f1oeq1d 6777	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑥–1-1-onto→(ω ↑o 𝑤)))
29		f1oeq2 6771	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑏):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
30	28, 29	bitrd 279	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
31	30	rexbidv 3162	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
32	26, 31	bitrid 283	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
33	23, 32	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ((ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ↔ (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))))
34	33	cbvralvw 3216	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ↔ ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
35	22, 34	sylib 218	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
36		oveq2 7376	. . . . . . . . 9 ⊢ (𝑏 = 𝑧 → (ω ↑o 𝑏) = (ω ↑o 𝑧))
37	36	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
38	37	cnveqi 5831	. . . . . . 7 ⊢ ◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = ◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
39	38	fveq1i 6843	. . . . . 6 ⊢ (◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏))‘ran (𝑛‘𝑏)) = (◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))‘ran (𝑛‘𝑏))
40		2on 8420	. . . . . . . . . 10 ⊢ 2o ∈ On
41		peano1 7841	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
42		oen0 8524	. . . . . . . . . . 11 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
43	41, 42	mpan2 692	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → ∅ ∈ (ω ↑o 2o))
44	4, 40, 43	mp2an 693	. . . . . . . . 9 ⊢ ∅ ∈ (ω ↑o 2o)
45		eqid 2737	. . . . . . . . . 10 ⊢ (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))
46	45	fveqf1o 7258	. . . . . . . . 9 ⊢ ((𝑓:(ω ↑o 2o)–1-1-onto→ω ∧ ∅ ∈ (ω ↑o 2o) ∧ ∅ ∈ ω) → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω ∧ ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅))
47	44, 41, 46	mp3an23 1456	. . . . . . . 8 ⊢ (𝑓:(ω ↑o 2o)–1-1-onto→ω → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω ∧ ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅))
48	47	ad2antll 730	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω ∧ ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅))
49	48	simpld 494	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω)
50	48	simprd 495	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅)
51	21, 35, 39, 49, 50	infxpenc2lem3 9943	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
52	51	ex 412	. . . 4 ⊢ (𝐴 ∈ On → ((∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
53	52	exlimdvv 1936	. . 3 ⊢ (𝐴 ∈ On → (∃𝑛∃𝑓(∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
54	20, 53	biimtrrid 243	. 2 ⊢ (𝐴 ∈ On → ((∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
55	1, 19, 54	mp2and 700	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  {cpr 4584  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   I cid 5526   × cxp 5630  ^◡ccnv 5631  ran crn 5633   ↾ cres 5634   ∘ ccom 5636  Oncon0 6325  suc csuc 6327  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  ωcom 7818  1_oc1o 8400  2_oc2o 8401   ·_o comu 8405   ↑_o coe 8406   ≈ cen 8892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-seqom 8389  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412  df-oexp 8413  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-cnf 9583  df-card 9863

This theorem is referenced by:  pwfseq  10587