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Mirrors > Home > MPE Home > Th. List > fnseqom | Structured version Visualization version GIF version |
Description: An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
seqom.a | ⊢ 𝐺 = seqω(𝐹, 𝐼) |
Ref | Expression |
---|---|
fnseqom | ⊢ 𝐺 Fn ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqomlem0 8163 | . . 3 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | |
2 | 1 | seqomlem2 8165 | . 2 ⊢ (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) Fn ω |
3 | seqom.a | . . . 4 ⊢ 𝐺 = seqω(𝐹, 𝐼) | |
4 | df-seqom 8162 | . . . 4 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | |
5 | 3, 4 | eqtri 2759 | . . 3 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) |
6 | 5 | fneq1i 6454 | . 2 ⊢ (𝐺 Fn ω ↔ (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) Fn ω) |
7 | 2, 6 | mpbir 234 | 1 ⊢ 𝐺 Fn ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 Vcvv 3398 ∅c0 4223 〈cop 4533 I cid 5439 “ cima 5539 suc csuc 6193 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 ωcom 7622 reccrdg 8123 seqωcseqom 8161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-seqom 8162 |
This theorem is referenced by: cantnfvalf 9258 fin23lem16 9914 fin23lem20 9916 fin23lem17 9917 fin23lem21 9918 fin23lem31 9922 |
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