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Mirrors > Home > MPE Home > Th. List > seqom0g | Structured version Visualization version GIF version |
Description: Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by AV, 17-Sep-2021.) |
Ref | Expression |
---|---|
seqom.a | ⊢ 𝐺 = seq𝜔(𝐹, 𝐼) |
Ref | Expression |
---|---|
seqom0g | ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqom.a | . . . . 5 ⊢ 𝐺 = seq𝜔(𝐹, 𝐼) | |
2 | df-seqom 7935 | . . . . 5 ⊢ seq𝜔(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | |
3 | 1, 2 | eqtri 2819 | . . . 4 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) |
4 | 3 | fveq1i 6539 | . . 3 ⊢ (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) |
5 | seqomlem0 7936 | . . . 4 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | |
6 | 5 | seqomlem3 7939 | . . 3 ⊢ ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) = ( I ‘𝐼) |
7 | 4, 6 | eqtri 2819 | . 2 ⊢ (𝐺‘∅) = ( I ‘𝐼) |
8 | fvi 6607 | . 2 ⊢ (𝐼 ∈ 𝑉 → ( I ‘𝐼) = 𝐼) | |
9 | 7, 8 | syl5eq 2843 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∅c0 4211 〈cop 4478 I cid 5347 “ cima 5446 suc csuc 6068 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 ωcom 7436 reccrdg 7897 seq𝜔cseqom 7934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-seqom 7935 |
This theorem is referenced by: cantnfvalf 8974 cantnfval2 8978 cantnflt 8981 cantnff 8983 cantnf0 8984 cantnfp1lem3 8989 cantnf 9002 cnfcom 9009 fseqenlem1 9296 fin23lem14 9601 fin23lem16 9603 |
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