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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 7590 | . . . . . . 7 ⊢ ∅ ∈ ω | |
2 | fvres 6682 | . . . . . . 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 6664 | . . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉)‘∅) |
6 | opex 5347 | . . . . . . 7 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ V | |
7 | 6 | rdg0 8046 | . . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉)‘∅) = 〈∅, ( I ‘𝐼)〉 |
8 | 3, 5, 7 | 3eqtri 2845 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = 〈∅, ( I ‘𝐼)〉 |
9 | frfnom 8059 | . . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) Fn ω | |
10 | 4 | reseq1i 5842 | . . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) |
11 | 10 | fneq1i 6443 | . . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) Fn ω) |
12 | 9, 11 | mpbir 232 | . . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω |
13 | fnfvelrn 6840 | . . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)) | |
14 | 12, 1, 13 | mp2an 688 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω) |
15 | 8, 14 | eqeltrri 2907 | . . . 4 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ ran (𝑄 ↾ ω) |
16 | df-ima 5561 | . . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω) | |
17 | 15, 16 | eleqtrri 2909 | . . 3 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ (𝑄 “ ω) |
18 | df-br 5058 | . . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ 〈∅, ( I ‘𝐼)〉 ∈ (𝑄 “ ω)) | |
19 | 17, 18 | mpbir 232 | . 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼) |
20 | 4 | seqomlem2 8076 | . . 3 ⊢ (𝑄 “ ω) Fn ω |
21 | fnbrfvb 6711 | . . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))) | |
22 | 20, 1, 21 | mp2an 688 | . 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)) |
23 | 19, 22 | mpbir 232 | 1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 class class class wbr 5057 I cid 5452 ran crn 5549 ↾ cres 5550 “ cima 5551 suc csuc 6186 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ωcom 7569 reccrdg 8034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 |
This theorem is referenced by: seqom0g 8081 |
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