Theorem infxpenc2lem1 10003

Description: Lemma for infxpenc2 10006. (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 3263	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
3	2	impr 459	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))
4		simpr 489	. . 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 7419	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑o 𝑥) = (ω ↑o 𝑤))
7		eqid 2769	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) = (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))
8		ovex 7444	. . . . . . . . . 10 ⊢ (ω ↑o 𝑤) ∈ V
9	6, 7, 8	fvmpt 6990	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1o) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = (ω ↑o 𝑤))
10	9	ad2antrl 740	. . . . . . . 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 6829	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤))
12	11	ad2antll 741	. . . . . . . . 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 6796	. . . . . . . . 9 ⊢ ((𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤) → ran (𝑛‘𝑏) = (ω ↑o 𝑤))
14	12, 13	syl 18	. . . . . . . 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 2807	. . . . . . 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 7444	. . . . . . . . . . 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 9615	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19		1onn 8626	. . . . . . . . . . . . 13 ⊢ 1o ∈ ω
20		ondif2 8487	. . . . . . . . . . . . 13 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
21	18, 19, 20	mpbir2an 723	. . . . . . . . . . . 12 ⊢ ω ∈ (On ∖ 2o)
22		eldifi 4093	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1o) → 𝑥 ∈ On)
23	22	ad2antrl 740	. . . . . . . . . . . 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)
24		eldifi 4093	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1o) → 𝑦 ∈ On)
25	24	ad2antll 741	. . . . . . . . . . . 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)
26		oecan 8575	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
27	21, 23, 25, 26	mp3an2i 1492	. . . . . . . . . . 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))
28	27	ex 417	. . . . . . . . . 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)))
29	17, 28	dom2lem 8989	. . . . . . . . 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)
30		f1f1orn 6833	. . . . . . . . 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)))
31	29, 30	syl 18	. . . . . . . 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)))
32		simprl 782	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑤 ∈ (On ∖ 1o))
33		f1ocnvfv 7277	. . . . . . . 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))
34	31, 32, 33	syl2anc 595	. . . . . . 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))
35	15, 34	mpd 16	. . . . . 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)
36	5, 35	eqtrid 2816	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑊 = 𝑤)
37	36	eleq1d 2854	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑊 ∈ (On ∖ 1o) ↔ 𝑤 ∈ (On ∖ 1o)))
38	36	oveq2d 7427	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (ω ↑o 𝑊) = (ω ↑o 𝑤))
39	38	f1oeq3d 6818	. . . 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)))
40	37, 39	anbi12d 643	. . 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))))
41	4, 40	mpbird 260	. 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)))
42	3, 41	rexlimddv 3178	1 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913   ↦ cmpt 5196  ^◡ccnv 5661  ran crn 5663  Oncon0 6361  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  ωcom 7862  1_oc1o 8446  2_oc2o 8447   ↑_o coe 8452

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-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-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-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-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-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-omul 8458  df-oexp 8459

This theorem is referenced by:  infxpenc2lem2  10004