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| 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 8488 | . . . . 5 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | |
| 3 | 1, 2 | eqtri 2765 | . . . 4 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | 
| 4 | 3 | fveq1i 6907 | . . 3 ⊢ (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) | 
| 5 | seqomlem0 8489 | . . . 4 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | |
| 6 | 5 | seqomlem3 8492 | . . 3 ⊢ ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) = ( I ‘𝐼) | 
| 7 | 4, 6 | eqtri 2765 | . 2 ⊢ (𝐺‘∅) = ( I ‘𝐼) | 
| 8 | fvi 6985 | . 2 ⊢ (𝐼 ∈ 𝑉 → ( I ‘𝐼) = 𝐼) | |
| 9 | 7, 8 | eqtrid 2789 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 I cid 5577 “ cima 5688 suc csuc 6386 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8449 seqωcseqom 8487 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 | 
| This theorem is referenced by: cantnfvalf 9705 cantnfval2 9709 cantnflt 9712 cantnff 9714 cantnf0 9715 cantnfp1lem3 9720 cantnf 9733 cnfcom 9740 fseqenlem1 10064 fin23lem14 10373 fin23lem16 10375 | 
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