Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvovco | Structured version Visualization version GIF version |
Description: Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
fvovco.1 | ⊢ (𝜑 → 𝐹:𝑋⟶(𝑉 × 𝑊)) |
fvovco.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
Ref | Expression |
---|---|
fvovco | ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvovco.1 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(𝑉 × 𝑊)) | |
2 | fvovco.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
3 | 1, 2 | ffvelrnd 6852 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊)) |
4 | 1st2nd2 7728 | . . . 4 ⊢ ((𝐹‘𝑌) ∈ (𝑉 × 𝑊) → (𝐹‘𝑌) = 〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) = 〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉) |
6 | 5 | fveq2d 6674 | . 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉)) |
7 | fvco3 6760 | . . 3 ⊢ ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌 ∈ 𝑋) → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌))) | |
8 | 1, 2, 7 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌))) |
9 | df-ov 7159 | . . 3 ⊢ ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌))) = (𝑂‘〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉) | |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌))) = (𝑂‘〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉)) |
11 | 6, 8, 10 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: cnmetcoval 41485 volicoff 42300 voliooicof 42301 hoissre 42846 hoiprodcl 42849 hoicvr 42850 hoicvrrex 42858 ovn0lem 42867 ovnhoilem1 42903 ovnhoilem2 42904 hoicoto2 42907 ovnlecvr2 42912 ovncvr2 42913 ovolval2lem 42945 ovolval5lem3 42956 |
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