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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 8487 | . . . . 5 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) | |
3 | 1, 2 | eqtri 2763 | . . . 4 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω) |
4 | 3 | fveq1i 6908 | . . 3 ⊢ (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) |
5 | seqomlem0 8488 | . . . 4 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) | |
6 | 5 | seqomlem3 8491 | . . 3 ⊢ ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) “ ω)‘∅) = ( I ‘𝐼) |
7 | 4, 6 | eqtri 2763 | . 2 ⊢ (𝐺‘∅) = ( I ‘𝐼) |
8 | fvi 6985 | . 2 ⊢ (𝐼 ∈ 𝑉 → ( I ‘𝐼) = 𝐼) | |
9 | 7, 8 | eqtrid 2787 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 〈cop 4637 I cid 5582 “ cima 5692 suc csuc 6388 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 reccrdg 8448 seqωcseqom 8486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seqom 8487 |
This theorem is referenced by: cantnfvalf 9703 cantnfval2 9707 cantnflt 9710 cantnff 9712 cantnf0 9713 cantnfp1lem3 9718 cantnf 9731 cnfcom 9738 fseqenlem1 10062 fin23lem14 10371 fin23lem16 10373 |
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