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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 8396 | . . 3 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩) | |
2 | 1 | seqomlem2 8398 | . 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 8395 | . . . 4 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω) | |
5 | 3, 4 | eqtri 2765 | . . 3 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω) |
6 | 5 | fneq1i 6600 | . 2 ⊢ (𝐺 Fn ω ↔ (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω) Fn ω) |
7 | 2, 6 | mpbir 230 | 1 ⊢ 𝐺 Fn ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3446 ∅c0 4283 ⟨cop 4593 I cid 5531 “ cima 5637 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 ωcom 7803 reccrdg 8356 seqωcseqom 8394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-seqom 8395 |
This theorem is referenced by: cantnfvalf 9602 fin23lem16 10272 fin23lem20 10274 fin23lem17 10275 fin23lem21 10276 fin23lem31 10280 |
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