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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 8446 | . . . . 5 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω) | |
3 | 1, 2 | eqtri 2754 | . . . 4 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω) |
4 | 3 | fveq1i 6885 | . . 3 ⊢ (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅) |
5 | seqomlem0 8447 | . . . 4 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩) | |
6 | 5 | seqomlem3 8450 | . . 3 ⊢ ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅) = ( I ‘𝐼) |
7 | 4, 6 | eqtri 2754 | . 2 ⊢ (𝐺‘∅) = ( I ‘𝐼) |
8 | fvi 6960 | . 2 ⊢ (𝐼 ∈ 𝑉 → ( I ‘𝐼) = 𝐼) | |
9 | 7, 8 | eqtrid 2778 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∅c0 4317 ⟨cop 4629 I cid 5566 “ cima 5672 suc csuc 6359 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 ωcom 7851 reccrdg 8407 seqωcseqom 8445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-seqom 8446 |
This theorem is referenced by: cantnfvalf 9659 cantnfval2 9663 cantnflt 9666 cantnff 9668 cantnf0 9669 cantnfp1lem3 9674 cantnf 9687 cnfcom 9694 fseqenlem1 10018 fin23lem14 10327 fin23lem16 10329 |
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