Theorem fin1a2lem5 10406

Description: Lemma for fin1a2 10417. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypothesis

Ref	Expression
fin1a2lem.b	⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))

Assertion

Ref	Expression
fin1a2lem5	⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

Proof of Theorem fin1a2lem5

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nneob 8644	. 2 ⊢ (𝐴 ∈ ω → (∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎) ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)))
2		fin1a2lem.b	. . . . . 6 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3	2	fin1a2lem4 10405	. . . . 5 ⊢ 𝐸:ω–1-1→ω
4		f1fn 6772	. . . . 5 ⊢ (𝐸:ω–1-1→ω → 𝐸 Fn ω)
5	3, 4	ax-mp 5	. . . 4 ⊢ 𝐸 Fn ω
6		fvelrnb 6938	. . . 4 ⊢ (𝐸 Fn ω → (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴))
7	5, 6	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴)
8		eqcom 2767	. . . . 5 ⊢ ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (𝐸‘𝑎))
9	2	fin1a2lem3 10404	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎))
10	9	eqeq2d 2771	. . . . 5 ⊢ (𝑎 ∈ ω → (𝐴 = (𝐸‘𝑎) ↔ 𝐴 = (2o ·o 𝑎)))
11	8, 10	bitrid 286	. . . 4 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (2o ·o 𝑎)))
12	11	rexbiia 3107	. . 3 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
13	7, 12	bitri 278	. 2 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
14		fvelrnb 6938	. . . . 5 ⊢ (𝐸 Fn ω → (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴))
15	5, 14	ax-mp 5	. . . 4 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴)
16		eqcom 2767	. . . . . 6 ⊢ ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸‘𝑎))
17	9	eqeq2d 2771	. . . . . 6 ⊢ (𝑎 ∈ ω → (suc 𝐴 = (𝐸‘𝑎) ↔ suc 𝐴 = (2o ·o 𝑎)))
18	16, 17	bitrid 286	. . . . 5 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (2o ·o 𝑎)))
19	18	rexbiia 3107	. . . 4 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
20	15, 19	bitri 278	. . 3 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
21	20	notbii 323	. 2 ⊢ (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
22	1, 13, 21	3bitr4g 317	1 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ↦ cmpt 5186  ran crn 5656  suc csuc 6359   Fn wfn 6528  –1-1→wf1 6530  ‘cfv 6533  (class class class)co 7413  ωcom 7862  2_oc2o 8449   ·_o comu 8453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-omul 8460

This theorem is used by:  fin1a2lem6  10407