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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddcbv | Structured version Visualization version GIF version |
Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.) |
Ref | Expression |
---|---|
dvhvaddval.a | ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) |
Ref | Expression |
---|---|
dvhvaddcbv | ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhvaddval.a | . 2 ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) | |
2 | fveq2 6191 | . . . . 5 ⊢ (𝑓 = ℎ → (1st ‘𝑓) = (1st ‘ℎ)) | |
3 | 2 | coeq1d 5283 | . . . 4 ⊢ (𝑓 = ℎ → ((1st ‘𝑓) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑔))) |
4 | fveq2 6191 | . . . . 5 ⊢ (𝑓 = ℎ → (2nd ‘𝑓) = (2nd ‘ℎ)) | |
5 | 4 | oveq1d 6665 | . . . 4 ⊢ (𝑓 = ℎ → ((2nd ‘𝑓) ⨣ (2nd ‘𝑔)) = ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))) |
6 | 3, 5 | opeq12d 4410 | . . 3 ⊢ (𝑓 = ℎ → 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉 = 〈((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))〉) |
7 | fveq2 6191 | . . . . 5 ⊢ (𝑔 = 𝑖 → (1st ‘𝑔) = (1st ‘𝑖)) | |
8 | 7 | coeq2d 5284 | . . . 4 ⊢ (𝑔 = 𝑖 → ((1st ‘ℎ) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑖))) |
9 | fveq2 6191 | . . . . 5 ⊢ (𝑔 = 𝑖 → (2nd ‘𝑔) = (2nd ‘𝑖)) | |
10 | 9 | oveq2d 6666 | . . . 4 ⊢ (𝑔 = 𝑖 → ((2nd ‘ℎ) ⨣ (2nd ‘𝑔)) = ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))) |
11 | 8, 10 | opeq12d 4410 | . . 3 ⊢ (𝑔 = 𝑖 → 〈((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))〉 = 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
12 | 6, 11 | cbvmpt2v 6735 | . 2 ⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
13 | 1, 12 | eqtri 2644 | 1 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 〈cop 4183 × cxp 5112 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1st c1st 7166 2nd c2nd 7167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-co 5123 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: dvhvaddval 36379 |
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