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 6844 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊)) |
4 | 1st2nd2 7717 | . . . 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 6667 | . 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘〈(1st ‘(𝐹‘𝑌)), (2nd ‘(𝐹‘𝑌))〉)) |
7 | fvco3 6753 | . . 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 7148 | . . 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 2863 | 1 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 〈cop 4563 × cxp 5546 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-1st 7678 df-2nd 7679 |
This theorem is referenced by: cnmetcoval 41341 volicoff 42157 voliooicof 42158 hoissre 42703 hoiprodcl 42706 hoicvr 42707 hoicvrrex 42715 ovn0lem 42724 ovnhoilem1 42760 ovnhoilem2 42761 hoicoto2 42764 ovnlecvr2 42769 ovncvr2 42770 ovolval2lem 42802 ovolval5lem3 42813 |
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