Theorem infxpenc2lem1 8842

Description: Lemma for infxpenc2 8845. (Contributed by Mario Carneiro, 30-May-2015.)

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))

Assertion

Ref	Expression
infxpenc2lem1	⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)))

Distinct variable groups:   𝑛,𝑏,𝑤,𝑥,𝐴   𝜑,𝑏,𝑤,𝑥   𝑤,𝑊,𝑥

Allowed substitution hints:   𝜑(𝑛)   𝑊(𝑛,𝑏)

Proof of Theorem infxpenc2lem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxpenc2.2	. . . 4 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
2	1	r19.21bi 2932	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
3	2	impr 649	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))
4		simpr 477	. . 3 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
5		infxpenc2.3	. . . . . 6 ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛‘𝑏))
6		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑤))
7		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)) = (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))
8		ovex 6678	. . . . . . . . . 10 ⊢ (ω ↑𝑜 𝑤) ∈ V
9	6, 7, 8	fvmpt 6282	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1𝑜) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = (ω ↑𝑜 𝑤))
10	9	ad2antrl 764	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = (ω ↑𝑜 𝑤))
11		f1ofo 6144	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑𝑜 𝑤))
12	11	ad2antll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑛‘𝑏):𝑏–onto→(ω ↑𝑜 𝑤))
13		forn 6118	. . . . . . . . 9 ⊢ ((𝑛‘𝑏):𝑏–onto→(ω ↑𝑜 𝑤) → ran (𝑛‘𝑏) = (ω ↑𝑜 𝑤))
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → ran (𝑛‘𝑏) = (ω ↑𝑜 𝑤))
15	10, 14	eqtr4d 2659	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = ran (𝑛‘𝑏))
16		ovex 6678	. . . . . . . . . . 11 ⊢ (ω ↑𝑜 𝑥) ∈ V
17	16	2a1i 12	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑥 ∈ (On ∖ 1𝑜) → (ω ↑𝑜 𝑥) ∈ V))
18		omelon 8543	. . . . . . . . . . . . . 14 ⊢ ω ∈ On
19		1onn 7719	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ ω
20		ondif2 7582	. . . . . . . . . . . . . 14 ⊢ (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
21	18, 19, 20	mpbir2an 955	. . . . . . . . . . . . 13 ⊢ ω ∈ (On ∖ 2𝑜)
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → ω ∈ (On ∖ 2𝑜))
23		eldifi 3732	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1𝑜) → 𝑥 ∈ On)
24	23	ad2antrl 764	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → 𝑥 ∈ On)
25		eldifi 3732	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1𝑜) → 𝑦 ∈ On)
26	25	ad2antll 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → 𝑦 ∈ On)
27		oecan 7669	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2𝑜) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦))
28	22, 24, 26, 27	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) ∧ (𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜))) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦))
29	28	ex 450	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑥 ∈ (On ∖ 1𝑜) ∧ 𝑦 ∈ (On ∖ 1𝑜)) → ((ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦) ↔ 𝑥 = 𝑦)))
30	17, 29	dom2lem 7995	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1→V)
31		f1f1orn 6148	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1→V → (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1-onto→ran (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)))
32	30, 31	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1-onto→ran (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)))
33		simprl 794	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → 𝑤 ∈ (On ∖ 1𝑜))
34		f1ocnvfv 6534	. . . . . . . 8 ⊢ (((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)):(On ∖ 1𝑜)–1-1-onto→ran (𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥)) ∧ 𝑤 ∈ (On ∖ 1𝑜)) → (((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = ran (𝑛‘𝑏) → (◡(𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛‘𝑏)) = 𝑤))
35	32, 33, 34	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘𝑤) = ran (𝑛‘𝑏) → (◡(𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛‘𝑏)) = 𝑤))
36	15, 35	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (◡(𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛‘𝑏)) = 𝑤)
37	5, 36	syl5eq 2668	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → 𝑊 = 𝑤)
38	37	eleq1d 2686	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑊 ∈ (On ∖ 1𝑜) ↔ 𝑤 ∈ (On ∖ 1𝑜)))
39	37	oveq2d 6666	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (ω ↑𝑜 𝑊) = (ω ↑𝑜 𝑤))
40		f1oeq3 6129	. . . . 5 ⊢ ((ω ↑𝑜 𝑊) = (ω ↑𝑜 𝑤) → ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
41	39, 40	syl 17	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤)))
42	38, 41	anbi12d 747	. . 3 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → ((𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)) ↔ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))))
43	4, 42	mpbird 247	. 2 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑤))) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)))
44	3, 43	rexlimddv 3035	1 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑𝑜 𝑊)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574   ↦ cmpt 4729  ^◡ccnv 5113  ran crn 5115  Oncon0 5723  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑜 coe 7559

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-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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  infxpenc2lem2  8843