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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 7881 | . . . . . . 7 ⊢ ∅ ∈ ω | |
2 | fvres 6910 | . . . . . . 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 6892 | . . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) |
6 | opex 5464 | . . . . . . 7 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V | |
7 | 6 | rdg0 8423 | . . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩ |
8 | 3, 5, 7 | 3eqtri 2764 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩ |
9 | frfnom 8437 | . . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω | |
10 | 4 | reseq1i 5977 | . . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) |
11 | 10 | fneq1i 6646 | . . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω) |
12 | 9, 11 | mpbir 230 | . . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω |
13 | fnfvelrn 7082 | . . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)) | |
14 | 12, 1, 13 | mp2an 690 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω) |
15 | 8, 14 | eqeltrri 2830 | . . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω) |
16 | df-ima 5689 | . . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω) | |
17 | 15, 16 | eleqtrri 2832 | . . 3 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω) |
18 | df-br 5149 | . . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)) | |
19 | 17, 18 | mpbir 230 | . 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼) |
20 | 4 | seqomlem2 8453 | . . 3 ⊢ (𝑄 “ ω) Fn ω |
21 | fnbrfvb 6944 | . . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))) | |
22 | 20, 1, 21 | mp2an 690 | . 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)) |
23 | 19, 22 | mpbir 230 | 1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 I cid 5573 ran crn 5677 ↾ cres 5678 “ cima 5679 suc csuc 6366 Fn wfn 6538 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 ωcom 7857 reccrdg 8411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 |
This theorem is referenced by: seqom0g 8458 |
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