Theorem fin1a2lem5 10357

Description: Lemma for fin1a2 10368. (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 8620	. 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 10356	. . . . 5 ⊢ 𝐸:ω–1-1→ω
4		f1fn 6757	. . . . 5 ⊢ (𝐸:ω–1-1→ω → 𝐸 Fn ω)
5	3, 4	ax-mp 5	. . . 4 ⊢ 𝐸 Fn ω
6		fvelrnb 6921	. . . 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 2736	. . . . 5 ⊢ ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (𝐸‘𝑎))
9	2	fin1a2lem3 10355	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎))
10	9	eqeq2d 2740	. . . . 5 ⊢ (𝑎 ∈ ω → (𝐴 = (𝐸‘𝑎) ↔ 𝐴 = (2o ·o 𝑎)))
11	8, 10	bitrid 283	. . . 4 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (2o ·o 𝑎)))
12	11	rexbiia 3074	. . 3 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
13	7, 12	bitri 275	. 2 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
14		fvelrnb 6921	. . . . 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 2736	. . . . . 6 ⊢ ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸‘𝑎))
17	9	eqeq2d 2740	. . . . . 6 ⊢ (𝑎 ∈ ω → (suc 𝐴 = (𝐸‘𝑎) ↔ suc 𝐴 = (2o ·o 𝑎)))
18	16, 17	bitrid 283	. . . . 5 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (2o ·o 𝑎)))
19	18	rexbiia 3074	. . . 4 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
20	15, 19	bitri 275	. . 3 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
21	20	notbii 320	. 2 ⊢ (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
22	1, 13, 21	3bitr4g 314	1 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ↦ cmpt 5188  ran crn 5639  suc csuc 6334   Fn wfn 6506  –1-1→wf1 6508  ‘cfv 6511  (class class class)co 7387  ωcom 7842  2_oc2o 8428   ·_o comu 8432

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-omul 8439

This theorem is referenced by:  fin1a2lem6  10358