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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 7437 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏)) | |
| 2 | 1 | opeq2d 4880 | . . . . 5 ⊢ (𝐴 = 𝐵 → 〈suc 𝑎, (𝑎𝐴𝑏)〉 = 〈suc 𝑎, (𝑎𝐵𝑏)〉) |
| 3 | 2 | mpoeq3dv 7512 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉)) |
| 4 | fveq2 6906 | . . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷)) | |
| 5 | 4 | opeq2d 4880 | . . . 4 ⊢ (𝐶 = 𝐷 → 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) |
| 6 | rdgeq12 8453 | . . . 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 596 | . . 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 6077 | . 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 8488 | . 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) | |
| 10 | df-seqom 8488 | . 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω) | |
| 11 | 8, 9, 10 | 3eqtr4g 2802 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 |
| This theorem is referenced by: cantnffval 9703 cantnfval 9708 cantnfres 9717 cnfcomlem 9739 cnfcom2 9742 fin23lem33 10385 |
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