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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 4885 | . . . . 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 6907 | . . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷)) | |
5 | 4 | opeq2d 4885 | . . . 4 ⊢ (𝐶 = 𝐷 → 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) |
6 | rdgeq12 8452 | . . . 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 6079 | . 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 8487 | . 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) | |
10 | df-seqom 8487 | . 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω) | |
11 | 8, 9, 10 | 3eqtr4g 2800 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 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-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seqom 8487 |
This theorem is referenced by: cantnffval 9701 cantnfval 9706 cantnfres 9715 cnfcomlem 9737 cnfcom2 9740 fin23lem33 10383 |
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