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Mirrors > Home > MPE Home > Th. List > seqomlem3 | Structured version Visualization version GIF version |
Description: Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
seqomlem.a | ⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) |
Ref | Expression |
---|---|
seqomlem3 | ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7251 | . . . . . . 7 ⊢ ∅ ∈ ω | |
2 | fvres 6369 | . . . . . . 7 ⊢ (∅ ∈ ω → ((𝑄 ↾ ω)‘∅) = (𝑄‘∅)) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ ((𝑄 ↾ ω)‘∅) = (𝑄‘∅) |
4 | seqomlem.a | . . . . . . 7 ⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) | |
5 | 4 | fveq1i 6354 | . . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉)‘∅) |
6 | opex 5081 | . . . . . . 7 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ V | |
7 | 6 | rdg0 7687 | . . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉)‘∅) = 〈∅, ( I ‘𝐼)〉 |
8 | 3, 5, 7 | 3eqtri 2786 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = 〈∅, ( I ‘𝐼)〉 |
9 | frfnom 7700 | . . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) Fn ω | |
10 | 4 | reseq1i 5547 | . . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) |
11 | 10 | fneq1i 6146 | . . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) Fn ω) |
12 | 9, 11 | mpbir 221 | . . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω |
13 | fnfvelrn 6520 | . . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)) | |
14 | 12, 1, 13 | mp2an 710 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω) |
15 | 8, 14 | eqeltrri 2836 | . . . 4 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ ran (𝑄 ↾ ω) |
16 | df-ima 5279 | . . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω) | |
17 | 15, 16 | eleqtrri 2838 | . . 3 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ (𝑄 “ ω) |
18 | df-br 4805 | . . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ 〈∅, ( I ‘𝐼)〉 ∈ (𝑄 “ ω)) | |
19 | 17, 18 | mpbir 221 | . 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼) |
20 | 4 | seqomlem2 7716 | . . 3 ⊢ (𝑄 “ ω) Fn ω |
21 | fnbrfvb 6398 | . . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))) | |
22 | 20, 1, 21 | mp2an 710 | . 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)) |
23 | 19, 22 | mpbir 221 | 1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∅c0 4058 〈cop 4327 class class class wbr 4804 I cid 5173 ran crn 5267 ↾ cres 5268 “ cima 5269 suc csuc 5886 Fn wfn 6044 ‘cfv 6049 (class class class)co 6814 ↦ cmpt2 6816 ωcom 7231 reccrdg 7675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 |
This theorem is referenced by: seqom0g 7721 |
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