Theorem infxpenc2lem1 9445

Description: Lemma for infxpenc2 9448. (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 3208	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
3	2	impr 457	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))
4		simpr 487	. . 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 7164	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑o 𝑥) = (ω ↑o 𝑤))
7		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) = (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))
8		ovex 7189	. . . . . . . . . 10 ⊢ (ω ↑o 𝑤) ∈ V
9	6, 7, 8	fvmpt 6768	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1o) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = (ω ↑o 𝑤))
10	9	ad2antrl 726	. . . . . . . 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 6622	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤))
12	11	ad2antll 727	. . . . . . . . 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 6593	. . . . . . . . 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 2859	. . . . . . 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 7189	. . . . . . . . . . 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 9109	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19		1onn 8265	. . . . . . . . . . . . 13 ⊢ 1o ∈ ω
20		ondif2 8127	. . . . . . . . . . . . 13 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
21	18, 19, 20	mpbir2an 709	. . . . . . . . . . . 12 ⊢ ω ∈ (On ∖ 2o)
22		eldifi 4103	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1o) → 𝑥 ∈ On)
23	22	ad2antrl 726	. . . . . . . . . . . 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 4103	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1o) → 𝑦 ∈ On)
25	24	ad2antll 727	. . . . . . . . . . . 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 8215	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
27	21, 23, 25, 26	mp3an2i 1462	. . . . . . . . . . 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 415	. . . . . . . . . 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 8549	. . . . . . . . 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 6626	. . . . . . . . 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 769	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑤 ∈ (On ∖ 1o))
33		f1ocnvfv 7035	. . . . . . . 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 586	. . . . . . 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	syl5eq 2868	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑊 = 𝑤)
37	36	eleq1d 2897	. . . 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 7172	. . . . 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 6612	. . . 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 259	. 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 3291	1 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936   ↦ cmpt 5146  ^◡ccnv 5554  ran crn 5556  Oncon0 6191  –1-1→wf1 6352  –onto→wfo 6353  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156  ωcom 7580  1_oc1o 8095  2_oc2o 8096   ↑_o coe 8101

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-omul 8107  df-oexp 8108

This theorem is referenced by:  infxpenc2lem2  9446