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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.) (Revise 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 7430 | . . . . 5 ⊢ seq𝜔(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | |
3 | 1, 2 | eqtri 2632 | . . . 4 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) |
4 | 3 | fveq1i 6104 | . . 3 ⊢ (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) |
5 | seqomlem0 7431 | . . . 4 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | |
6 | 5 | seqomlem3 7434 | . . 3 ⊢ ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) = ( I ‘𝐼) |
7 | 4, 6 | eqtri 2632 | . 2 ⊢ (𝐺‘∅) = ( I ‘𝐼) |
8 | fvi 6165 | . 2 ⊢ (𝐼 ∈ 𝑉 → ( I ‘𝐼) = 𝐼) | |
9 | 7, 8 | syl5eq 2656 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 I cid 4948 “ cima 5041 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 reccrdg 7392 seq𝜔cseqom 7429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 |
This theorem is referenced by: cantnfvalf 8445 cantnfval2 8449 cantnflt 8452 cantnff 8454 cantnf0 8455 cantnfp1lem3 8460 cantnf 8473 cnfcom 8480 fseqenlem1 8730 fin23lem14 9038 fin23lem16 9040 |
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