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Mirrors > Home > MPE Home > Th. List > seqomeq12 | Structured version Visualization version GIF version |
Description: Equality theorem for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
seqomeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7151 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏)) | |
2 | 1 | opeq2d 4802 | . . . . 5 ⊢ (𝐴 = 𝐵 → 〈suc 𝑎, (𝑎𝐴𝑏)〉 = 〈suc 𝑎, (𝑎𝐵𝑏)〉) |
3 | 2 | mpoeq3dv 7222 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉)) |
4 | fveq2 6663 | . . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷)) | |
5 | 4 | opeq2d 4802 | . . . 4 ⊢ (𝐶 = 𝐷 → 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) |
6 | rdgeq12 8038 | . . . 4 ⊢ (((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉) ∧ 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉)) | |
7 | 3, 5, 6 | syl2an 595 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) = rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉)) |
8 | 7 | imaeq1d 5921 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω)) |
9 | df-seqom 8073 | . 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) | |
10 | df-seqom 8073 | . 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω) | |
11 | 8, 9, 10 | 3eqtr4g 2878 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Vcvv 3492 ∅c0 4288 〈cop 4563 I cid 5452 “ cima 5551 suc csuc 6186 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ωcom 7569 reccrdg 8034 seqωcseqom 8072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-iota 6307 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seqom 8073 |
This theorem is referenced by: cantnffval 9114 cantnfval 9119 cantnfres 9128 cnfcomlem 9150 cnfcom2 9153 fin23lem33 9755 |
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