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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 5707 | . . . 4 ⊢ (𝑎 = 𝑐 → suc 𝑎 = suc 𝑐) | |
2 | oveq1 6556 | . . . 4 ⊢ (𝑎 = 𝑐 → (𝑎𝐹𝑏) = (𝑐𝐹𝑏)) | |
3 | 1, 2 | opeq12d 4348 | . . 3 ⊢ (𝑎 = 𝑐 → 〈suc 𝑎, (𝑎𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑏)〉) |
4 | oveq2 6557 | . . . 4 ⊢ (𝑏 = 𝑑 → (𝑐𝐹𝑏) = (𝑐𝐹𝑑)) | |
5 | 4 | opeq2d 4347 | . . 3 ⊢ (𝑏 = 𝑑 → 〈suc 𝑐, (𝑐𝐹𝑏)〉 = 〈suc 𝑐, (𝑐𝐹𝑑)〉) |
6 | 3, 5 | cbvmpt2v 6633 | . 2 ⊢ (𝑎 ∈ ω, 𝑏 ∈ V ↦ 〈suc 𝑎, (𝑎𝐹𝑏)〉) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ 〈suc 𝑐, (𝑐𝐹𝑑)〉) |
7 | rdgeq1 7394 | . 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 1475 Vcvv 3173 ∅c0 3874 〈cop 4131 I cid 4948 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 reccrdg 7392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-suc 5646 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 |
This theorem is referenced by: fnseqom 7437 seqom0g 7438 seqomsuc 7439 |
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