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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 8085 | . . 3 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | |
2 | 1 | seqomlem2 8087 | . 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 8084 | . . . 4 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | |
5 | 3, 4 | eqtri 2844 | . . 3 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) |
6 | 5 | fneq1i 6450 | . 2 ⊢ (𝐺 Fn ω ↔ (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) Fn ω) |
7 | 2, 6 | mpbir 233 | 1 ⊢ 𝐺 Fn ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3494 ∅c0 4291 〈cop 4573 I cid 5459 “ cima 5558 suc csuc 6193 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ωcom 7580 reccrdg 8045 seqωcseqom 8083 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seqom 8084 |
This theorem is referenced by: cantnfvalf 9128 fin23lem16 9757 fin23lem20 9759 fin23lem17 9760 fin23lem21 9761 fin23lem31 9765 |
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