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Mirrors > Home > MPE Home > Th. List > fin1a2lem5 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9120. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
fin1a2lem.b | ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥)) |
Ref | Expression |
---|---|
fin1a2lem5 | ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nneob 7619 | . 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 9108 | . . . . 5 ⊢ 𝐸:ω–1-1→ω |
4 | f1fn 6015 | . . . . 5 ⊢ (𝐸:ω–1-1→ω → 𝐸 Fn ω) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐸 Fn ω |
6 | fvelrnb 6153 | . . . 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 2617 | . . . . 5 ⊢ ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (𝐸‘𝑎)) | |
9 | 2 | fin1a2lem3 9107 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2𝑜 ·𝑜 𝑎)) |
10 | 9 | eqeq2d 2620 | . . . . 5 ⊢ (𝑎 ∈ ω → (𝐴 = (𝐸‘𝑎) ↔ 𝐴 = (2𝑜 ·𝑜 𝑎))) |
11 | 8, 10 | syl5bb 271 | . . . 4 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (2𝑜 ·𝑜 𝑎))) |
12 | 11 | rexbiia 3022 | . . 3 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑎)) |
13 | 7, 12 | bitri 263 | . 2 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑎)) |
14 | fvelrnb 6153 | . . . . 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 2617 | . . . . . 6 ⊢ ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸‘𝑎)) | |
17 | 9 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑎 ∈ ω → (suc 𝐴 = (𝐸‘𝑎) ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑎))) |
18 | 16, 17 | syl5bb 271 | . . . . 5 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑎))) |
19 | 18 | rexbiia 3022 | . . . 4 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎)) |
20 | 15, 19 | bitri 263 | . . 3 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎)) |
21 | 20 | notbii 309 | . 2 ⊢ (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎)) |
22 | 1, 13, 21 | 3bitr4g 302 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ↦ cmpt 4643 ran crn 5039 suc csuc 5642 Fn wfn 5799 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 ωcom 6957 2𝑜c2o 7441 ·𝑜 comu 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 |
This theorem is referenced by: fin1a2lem6 9110 |
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