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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 6977 | . . . . . . 7 ⊢ ∅ ∈ ω | |
2 | fvres 6117 | . . . . . . 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 6104 | . . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉)‘∅) |
6 | opex 4859 | . . . . . . 7 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ V | |
7 | 6 | rdg0 7404 | . . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉)‘∅) = 〈∅, ( I ‘𝐼)〉 |
8 | 3, 5, 7 | 3eqtri 2636 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = 〈∅, ( I ‘𝐼)〉 |
9 | frfnom 7417 | . . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) Fn ω | |
10 | 4 | reseq1i 5313 | . . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) |
11 | 10 | fneq1i 5899 | . . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) ↾ ω) Fn ω) |
12 | 9, 11 | mpbir 220 | . . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω |
13 | fnfvelrn 6264 | . . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)) | |
14 | 12, 1, 13 | mp2an 704 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω) |
15 | 8, 14 | eqeltrri 2685 | . . . 4 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ ran (𝑄 ↾ ω) |
16 | df-ima 5051 | . . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω) | |
17 | 15, 16 | eleqtrri 2687 | . . 3 ⊢ 〈∅, ( I ‘𝐼)〉 ∈ (𝑄 “ ω) |
18 | df-br 4584 | . . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ 〈∅, ( I ‘𝐼)〉 ∈ (𝑄 “ ω)) | |
19 | 17, 18 | mpbir 220 | . 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼) |
20 | 4 | seqomlem2 7433 | . . 3 ⊢ (𝑄 “ ω) Fn ω |
21 | fnbrfvb 6146 | . . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))) | |
22 | 20, 1, 21 | mp2an 704 | . 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)) |
23 | 19, 22 | mpbir 220 | 1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 class class class wbr 4583 I cid 4948 ran crn 5039 ↾ cres 5040 “ cima 5041 suc csuc 5642 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 reccrdg 7392 |
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-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-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 |
This theorem is referenced by: seqom0g 7438 |
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