Theorem infxpenc2 9934

Description: Existence form of infxpenc 9930. A "uniform" or "canonical" version of infxpen 9926, 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 9617	. 2 ⊢ (𝐴 ∈ On → ∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)))
2		df-2o 8398	. . . . . . . 8 ⊢ 2o = suc 1o
3	2	oveq2i 7369	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
4		omelon 9557	. . . . . . . 8 ⊢ ω ∈ On
5		1on 8409	. . . . . . . 8 ⊢ 1o ∈ On
6		oesuc 8454	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
7	4, 5, 6	mp2an 692	. . . . . . 7 ⊢ (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
8		oe1 8471	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
9	4, 8	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
10	9	oveq1i 7368	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
11	3, 7, 10	3eqtri 2763	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
12		omxpen 9009	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (ω ·o ω) ≈ (ω × ω))
13	4, 4, 12	mp2an 692	. . . . . 6 ⊢ (ω ·o ω) ≈ (ω × ω)
14	11, 13	eqbrtri 5119	. . . . 5 ⊢ (ω ↑o 2o) ≈ (ω × ω)
15		xpomen 9927	. . . . 5 ⊢ (ω × ω) ≈ ω
16	14, 15	entri 8947	. . . 4 ⊢ (ω ↑o 2o) ≈ ω
17	16	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (ω ↑o 2o) ≈ ω)
18		bren 8895	. . 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 1956	. . 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 770	. . . . . . 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 3960	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (ω ↑o 𝑦) = (ω ↑o 𝑤))
25	24	f1oeq3d 6771	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ (𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤)))
26	25	cbvrexvw 3215	. . . . . . . . . 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 6834	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝑛‘𝑥) = (𝑛‘𝑏))
28	27	f1oeq1d 6769	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑥–1-1-onto→(ω ↑o 𝑤)))
29		f1oeq2 6763	. . . . . . . . . . . 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 3160	. . . . . . . . . 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 3214	. . . . . . 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 7366	. . . . . . . . 9 ⊢ (𝑏 = 𝑧 → (ω ↑o 𝑏) = (ω ↑o 𝑧))
37	36	cbvmptv 5202	. . . . . . . 8 ⊢ (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
38	37	cnveqi 5823	. . . . . . 7 ⊢ ◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = ◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
39	38	fveq1i 6835	. . . . . 6 ⊢ (◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏))‘ran (𝑛‘𝑏)) = (◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))‘ran (𝑛‘𝑏))
40		2on 8410	. . . . . . . . . 10 ⊢ 2o ∈ On
41		peano1 7831	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
42		oen0 8514	. . . . . . . . . . 11 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
43	41, 42	mpan2 691	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → ∅ ∈ (ω ↑o 2o))
44	4, 40, 43	mp2an 692	. . . . . . . . 9 ⊢ ∅ ∈ (ω ↑o 2o)
45		eqid 2736	. . . . . . . . . 10 ⊢ (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))
46	45	fveqf1o 7248	. . . . . . . . 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 1455	. . . . . . . 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 729	. . . . . . 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 9933	. . . . 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 1935	. . 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 699	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {cpr 4582  ⟨cop 4586   class class class wbr 5098   ↦ cmpt 5179   I cid 5518   × cxp 5622  ^◡ccnv 5623  ran crn 5625   ↾ cres 5626   ∘ ccom 5628  Oncon0 6317  suc csuc 6319  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  ωcom 7808  1_oc1o 8390  2_oc2o 8391   ·_o comu 8395   ↑_o coe 8396   ≈ cen 8882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seqom 8379  df-1o 8397  df-2o 8398  df-oadd 8401  df-omul 8402  df-oexp 8403  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fsupp 9267  df-oi 9417  df-cnf 9573  df-card 9853

This theorem is referenced by:  pwfseq  10577