Theorem infxpenc2lem1 9979

Description: Lemma for infxpenc2 9982. (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 3230	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
3	2	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))
4		simpr 484	. . 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 7398	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑o 𝑥) = (ω ↑o 𝑤))
7		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) = (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))
8		ovex 7423	. . . . . . . . . 10 ⊢ (ω ↑o 𝑤) ∈ V
9	6, 7, 8	fvmpt 6971	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1o) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = (ω ↑o 𝑤))
10	9	ad2antrl 728	. . . . . . . 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 6810	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤))
12	11	ad2antll 729	. . . . . . . . 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 6778	. . . . . . . . 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 2768	. . . . . . 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 7423	. . . . . . . . . . 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 9606	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19		1onn 8607	. . . . . . . . . . . . 13 ⊢ 1o ∈ ω
20		ondif2 8469	. . . . . . . . . . . . 13 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
21	18, 19, 20	mpbir2an 711	. . . . . . . . . . . 12 ⊢ ω ∈ (On ∖ 2o)
22		eldifi 4097	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1o) → 𝑥 ∈ On)
23	22	ad2antrl 728	. . . . . . . . . . . 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 4097	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1o) → 𝑦 ∈ On)
25	24	ad2antll 729	. . . . . . . . . . . 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 8556	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
27	21, 23, 25, 26	mp3an2i 1468	. . . . . . . . . . 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 412	. . . . . . . . . 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 8966	. . . . . . . . 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 6814	. . . . . . . . 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 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)))
32		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑤 ∈ (On ∖ 1o))
33		f1ocnvfv 7256	. . . . . . . 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 584	. . . . . . 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 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)
36	5, 35	eqtrid 2777	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑊 = 𝑤)
37	36	eleq1d 2814	. . . 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 7406	. . . . 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 6800	. . . 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 632	. . 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 257	. 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 3141	1 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917   ↦ cmpt 5191  ^◡ccnv 5640  ran crn 5642  Oncon0 6335  –1-1→wf1 6511  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  ωcom 7845  1_oc1o 8430  2_oc2o 8431   ↑_o coe 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443

This theorem is referenced by:  infxpenc2lem2  9980