Theorem infxpenc2 8845

Description: Existence form of infxpenc 8841. A "uniform" or "canonical" version of infxpen 8837, 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 8603	. 2 ⊢ (𝐴 ∈ On → ∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑦)))
2		df-2o 7561	. . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜
3	2	oveq2i 6661	. . . . . . 7 ⊢ (ω ↑𝑜 2𝑜) = (ω ↑𝑜 suc 1𝑜)
4		omelon 8543	. . . . . . . 8 ⊢ ω ∈ On
5		1on 7567	. . . . . . . 8 ⊢ 1𝑜 ∈ On
6		oesuc 7607	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω))
7	4, 5, 6	mp2an 708	. . . . . . 7 ⊢ (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω)
8		oe1 7624	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
9	4, 8	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑𝑜 1𝑜) = ω
10	9	oveq1i 6660	. . . . . . 7 ⊢ ((ω ↑𝑜 1𝑜) ·𝑜 ω) = (ω ·𝑜 ω)
11	3, 7, 10	3eqtri 2648	. . . . . 6 ⊢ (ω ↑𝑜 2𝑜) = (ω ·𝑜 ω)
12		omxpen 8062	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (ω ·𝑜 ω) ≈ (ω × ω))
13	4, 4, 12	mp2an 708	. . . . . 6 ⊢ (ω ·𝑜 ω) ≈ (ω × ω)
14	11, 13	eqbrtri 4674	. . . . 5 ⊢ (ω ↑𝑜 2𝑜) ≈ (ω × ω)
15		xpomen 8838	. . . . 5 ⊢ (ω × ω) ≈ ω
16	14, 15	entri 8010	. . . 4 ⊢ (ω ↑𝑜 2𝑜) ≈ ω
17	16	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (ω ↑𝑜 2𝑜) ≈ ω)
18		bren 7964	. . 3 ⊢ ((ω ↑𝑜 2𝑜) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
19	17, 18	sylib 208	. 2 ⊢ (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
20		eeanv 2182	. . 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 473	. . . . . 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 794	. . . . . . 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 3627	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤))
25		f1oeq3 6129	. . . . . . . . . . . 12 ⊢ ((ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤) → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑦) ↔ (𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑤)))
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑦) ↔ (𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑤)))
27	26	cbvrexv 3172	. . . . . . . . . 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))
28		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝑛‘𝑥) = (𝑛‘𝑏))
29		f1oeq1 6127	. . . . . . . . . . . . 13 ⊢ ((𝑛‘𝑥) = (𝑛‘𝑏) → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛‘𝑏):𝑥–1-1-onto→(ω ↑𝑜 𝑤)))
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛‘𝑏):𝑥–1-1-onto→(ω ↑𝑜 𝑤)))
31		f1oeq2 6128	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑏):𝑥–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
32	30, 31	bitrd 268	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
33	32	rexbidv 3052	. . . . . . . . . 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)))
34	27, 33	syl5bb 272	. . . . . . . . 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)))
35	23, 34	imbi12d 334	. . . . . . . 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))))
36	35	cbvralv 3171	. . . . . . 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)))
37	22, 36	sylib 208	. . . . . 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)))
38		oveq2 6658	. . . . . . . . 9 ⊢ (𝑏 = 𝑧 → (ω ↑𝑜 𝑏) = (ω ↑𝑜 𝑧))
39	38	cbvmptv 4750	. . . . . . . 8 ⊢ (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
40	39	cnveqi 5297	. . . . . . 7 ⊢ ◡(𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = ◡(𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
41	40	fveq1i 6192	. . . . . 6 ⊢ (◡(𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏))‘ran (𝑛‘𝑏)) = (◡(𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))‘ran (𝑛‘𝑏))
42		2on 7568	. . . . . . . . . 10 ⊢ 2𝑜 ∈ On
43		peano1 7085	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
44		oen0 7666	. . . . . . . . . . 11 ⊢ (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
45	43, 44	mpan2 707	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2𝑜 ∈ On) → ∅ ∈ (ω ↑𝑜 2𝑜))
46	4, 42, 45	mp2an 708	. . . . . . . . 9 ⊢ ∅ ∈ (ω ↑𝑜 2𝑜)
47		eqid 2622	. . . . . . . . . 10 ⊢ (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))
48	47	fveqf1o 6557	. . . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
49	46, 43, 48	mp3an23 1416	. . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
50	49	ad2antll 765	. . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
51	50	simpld 475	. . . . . 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)
52	50	simprd 479	. . . . . 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‘∅), ∅⟩}))‘∅) = ∅)
53	21, 37, 41, 51, 52	infxpenc2lem3 8844	. . . . 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))
54	53	ex 450	. . . 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)))
55	54	exlimdvv 1862	. . 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)))
56	20, 55	syl5bir 233	. 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)))
57	1, 19, 56	mp2and 715	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {cpr 4179  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   I cid 5023   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   ∘ ccom 5118  Oncon0 5723  suc csuc 5725  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ·_𝑜 comu 7558   ↑_𝑜 coe 7559   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559  df-card 8765

This theorem is referenced by:  pwfseq  9486