Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7795 | . . . . . . 7 ⊢ ∅ ∈ ω | |
2 | fvres 6838 | . . . . . . 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 6820 | . . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) |
6 | opex 5403 | . . . . . . 7 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V | |
7 | 6 | rdg0 8314 | . . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩ |
8 | 3, 5, 7 | 3eqtri 2768 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩ |
9 | frfnom 8328 | . . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω | |
10 | 4 | reseq1i 5913 | . . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) |
11 | 10 | fneq1i 6576 | . . . . . . 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 7008 | . . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)) | |
14 | 12, 1, 13 | mp2an 689 | . . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω) |
15 | 8, 14 | eqeltrri 2834 | . . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω) |
16 | df-ima 5627 | . . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω) | |
17 | 15, 16 | eleqtrri 2836 | . . 3 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω) |
18 | df-br 5090 | . . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)) | |
19 | 17, 18 | mpbir 230 | . 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼) |
20 | 4 | seqomlem2 8344 | . . 3 ⊢ (𝑄 “ ω) Fn ω |
21 | fnbrfvb 6872 | . . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))) | |
22 | 20, 1, 21 | mp2an 689 | . 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 1540 ∈ wcel 2105 Vcvv 3441 ∅c0 4268 ⟨cop 4578 class class class wbr 5089 I cid 5511 ran crn 5615 ↾ cres 5616 “ cima 5617 suc csuc 6298 Fn wfn 6468 ‘cfv 6473 (class class class)co 7329 ∈ cmpo 7331 ωcom 7772 reccrdg 8302 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pr 5369 ax-un 7642 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 |
This theorem is referenced by: seqom0g 8349 |
Copyright terms: Public domain | W3C validator |