![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊)) |
4 | 1st2nd2 8052 | . . . 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 6911 | . 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉)) |
7 | fvco3 7008 | . . 3 ⊢ ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌 ∈ 𝑋) → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌))) | |
8 | 1, 2, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌))) |
9 | df-ov 7434 | . . 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 2785 | 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 2106 〈cop 4637 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: cnmetcoval 45145 volicoff 45951 voliooicof 45952 hoissre 46500 hoiprodcl 46503 hoicvr 46504 hoicvrrex 46512 ovn0lem 46521 ovnhoilem1 46557 ovnhoilem2 46558 hoicoto2 46561 ovnlecvr2 46566 ovncvr2 46567 ovolval2lem 46599 ovolval5lem3 46610 |
Copyright terms: Public domain | W3C validator |