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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 7164 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑎𝐴𝑏) = (𝑎𝐵𝑏)) | |
2 | 1 | opeq2d 4812 | . . . . 5 ⊢ (𝐴 = 𝐵 → 〈suc 𝑎, (𝑎𝐴𝑏)〉 = 〈suc 𝑎, (𝑎𝐵𝑏)〉) |
3 | 2 | mpoeq3dv 7235 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉) = (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉)) |
4 | fveq2 6672 | . . . . 5 ⊢ (𝐶 = 𝐷 → ( I ‘𝐶) = ( I ‘𝐷)) | |
5 | 4 | opeq2d 4812 | . . . 4 ⊢ (𝐶 = 𝐷 → 〈∅, ( I ‘𝐶)〉 = 〈∅, ( I ‘𝐷)〉) |
6 | rdgeq12 8051 | . . . 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 597 | . . 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 5930 | . 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 8086 | . 2 ⊢ seqω(𝐴, 𝐶) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐴𝑏)〉), 〈∅, ( I ‘𝐶)〉) “ ω) | |
10 | df-seqom 8086 | . 2 ⊢ seqω(𝐵, 𝐷) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐵𝑏)〉), 〈∅, ( I ‘𝐷)〉) “ ω) | |
11 | 8, 9, 10 | 3eqtr4g 2883 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Vcvv 3496 ∅c0 4293 〈cop 4575 I cid 5461 “ cima 5560 suc csuc 6195 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ωcom 7582 reccrdg 8047 seqωcseqom 8085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seqom 8086 |
This theorem is referenced by: cantnffval 9128 cantnfval 9133 cantnfres 9142 cnfcomlem 9164 cnfcom2 9167 fin23lem33 9769 |
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