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| Mirrors > Home > MPE Home > Th. List > opco2 | Structured version Visualization version GIF version | ||
| Description: Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.) |
| Ref | Expression |
|---|---|
| opco1.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| opco1.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| Ref | Expression |
|---|---|
| opco2 | ⊢ (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ov 7434 | . . 3 ⊢ (𝐴(𝐹 ∘ 2nd )𝐵) = ((𝐹 ∘ 2nd )‘〈𝐴, 𝐵〉) | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = ((𝐹 ∘ 2nd )‘〈𝐴, 𝐵〉)) |
| 3 | fo2nd 8035 | . . . 4 ⊢ 2nd :V–onto→V | |
| 4 | fof 6820 | . . . 4 ⊢ (2nd :V–onto→V → 2nd :V⟶V) | |
| 5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝜑 → 2nd :V⟶V) |
| 6 | opex 5469 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ V) |
| 8 | 5, 7 | fvco3d 7009 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 2nd )‘〈𝐴, 𝐵〉) = (𝐹‘(2nd ‘〈𝐴, 𝐵〉))) |
| 9 | opco1.exa | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 10 | opco1.exb | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 11 | op2ndg 8027 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
| 12 | 9, 10, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
| 13 | 12 | fveq2d 6910 | . 2 ⊢ (𝜑 → (𝐹‘(2nd ‘〈𝐴, 𝐵〉)) = (𝐹‘𝐵)) |
| 14 | 2, 8, 13 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ∘ ccom 5689 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-2nd 8015 |
| This theorem is referenced by: dfwrecsOLD 8338 wfr2a 8374 dfrecs3 8412 |
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