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Mirrors > Home > MPE Home > Th. List > seqomlem0 | Structured version Visualization version GIF version |
Description: Lemma for seqω. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
seqomlem0 | ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceq 6250 | . . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐) | |
2 | oveq1 7157 | . . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏)) | |
3 | 1, 2 | opeq12d 4804 | . . 3 ⊢ (𝑎 = 𝑐 → 〈suc 𝑎, (𝑎𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑏)〉) |
4 | oveq2 7158 | . . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑)) | |
5 | 4 | opeq2d 4803 | . . 3 ⊢ (𝑏 = 𝑑 → 〈suc 𝑐, (𝑐𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑑)〉) |
6 | 3, 5 | cbvmpov 7243 | . 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉) |
7 | rdgeq1 8041 | . 2 ⊢ ((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉) → rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉)) | |
8 | 6, 7 | ax-mp 5 | 1 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉), 〈∅, ( I ‘𝐼)〉) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉), 〈∅, ( I ‘𝐼)〉) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3494 ∅c0 4290 〈cop 4566 I cid 5453 suc csuc 6187 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ωcom 7574 reccrdg 8039 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-suc 6191 df-iota 6308 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-wrecs 7941 df-recs 8002 df-rdg 8040 |
This theorem is referenced by: fnseqom 8085 seqom0g 8086 seqomsuc 8087 |
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