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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 7272 | . . 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 7843 | . . . 4 ⊢ 2nd :V–onto→V | |
4 | fof 6682 | . . . 4 ⊢ (2nd :V–onto→V → 2nd :V⟶V) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝜑 → 2nd :V⟶V) |
6 | opex 5379 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ V) |
8 | 5, 7 | fvco3d 6862 | . 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 7835 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
12 | 9, 10, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
13 | 12 | fveq2d 6772 | . 2 ⊢ (𝜑 → (𝐹‘(2nd ‘〈𝐴, 𝐵〉)) = (𝐹‘𝐵)) |
14 | 2, 8, 13 | 3eqtrd 2782 | 1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3431 〈cop 4569 ∘ ccom 5590 ⟶wf 6424 –onto→wfo 6426 ‘cfv 6428 (class class class)co 7269 2nd c2nd 7821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-fo 6434 df-fv 6436 df-ov 7272 df-2nd 7823 |
This theorem is referenced by: dfwrecsOLD 8118 wfr2a 8154 dfrecs3 8192 |
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