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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 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊)) |
| 4 | 1st2nd2 8053 | . . . 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 6910 | . 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 2787 | 1 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1st ‘(𝐹‘𝑌))𝑂(2nd ‘(𝐹‘𝑌)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 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-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: cnmetcoval 45207 volicoff 46010 voliooicof 46011 hoissre 46559 hoiprodcl 46562 hoicvr 46563 hoicvrrex 46571 ovn0lem 46580 ovnhoilem1 46616 ovnhoilem2 46617 hoicoto2 46620 ovnlecvr2 46625 ovncvr2 46626 ovolval2lem 46658 ovolval5lem3 46669 |
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