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Mirrors > Home > MPE Home > Th. List > fin1a2lem5 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9837. (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 8279 | . 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 9825 | . . . . 5 ⊢ 𝐸:ω–1-1→ω |
4 | f1fn 6576 | . . . . 5 ⊢ (𝐸:ω–1-1→ω → 𝐸 Fn ω) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐸 Fn ω |
6 | fvelrnb 6726 | . . . 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 2828 | . . . . 5 ⊢ ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (𝐸‘𝑎)) | |
9 | 2 | fin1a2lem3 9824 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎)) |
10 | 9 | eqeq2d 2832 | . . . . 5 ⊢ (𝑎 ∈ ω → (𝐴 = (𝐸‘𝑎) ↔ 𝐴 = (2o ·o 𝑎))) |
11 | 8, 10 | syl5bb 285 | . . . 4 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (2o ·o 𝑎))) |
12 | 11 | rexbiia 3246 | . . 3 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎)) |
13 | 7, 12 | bitri 277 | . 2 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎)) |
14 | fvelrnb 6726 | . . . . 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 2828 | . . . . . 6 ⊢ ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸‘𝑎)) | |
17 | 9 | eqeq2d 2832 | . . . . . 6 ⊢ (𝑎 ∈ ω → (suc 𝐴 = (𝐸‘𝑎) ↔ suc 𝐴 = (2o ·o 𝑎))) |
18 | 16, 17 | syl5bb 285 | . . . . 5 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (2o ·o 𝑎))) |
19 | 18 | rexbiia 3246 | . . . 4 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)) |
20 | 15, 19 | bitri 277 | . . 3 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)) |
21 | 20 | notbii 322 | . 2 ⊢ (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)) |
22 | 1, 13, 21 | 3bitr4g 316 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ↦ cmpt 5146 ran crn 5556 suc csuc 6193 Fn wfn 6350 –1-1→wf1 6352 ‘cfv 6355 (class class class)co 7156 ωcom 7580 2oc2o 8096 ·o comu 8100 |
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 |
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 |
This theorem is referenced by: fin1a2lem6 9827 |
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