fvovco - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fvovco		Structured version Visualization version GIF version

Theorem fvovco 43535

Description: Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
fvovco.1	⊢ (𝜑 → 𝐹:𝑋⟶(𝑉 × 𝑊))
fvovco.2	⊢ (𝜑 → 𝑌 ∈ 𝑋)

**Assertion**
Ref	Expression
fvovco	⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1^st ‘(𝐹‘𝑌))𝑂(2^nd ‘(𝐹‘𝑌))))

**Proof of Theorem fvovco**
Step	Hyp	Ref	Expression
1		fvovco.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(𝑉 × 𝑊))
2		fvovco.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
3	1, 2	ffvelcdmd 7041	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊))
4		1st2nd2 7965	. . . 4 ⊢ ((𝐹‘𝑌) ∈ (𝑉 × 𝑊) → (𝐹‘𝑌) = ⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) = ⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩)
6	5	fveq2d 6851	. 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩))
7		fvco3 6945	. . 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 7365	. . 3 ⊢ ((1^st ‘(𝐹‘𝑌))𝑂(2^nd ‘(𝐹‘𝑌))) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩)
10	9	a1i 11	. 2 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑌))𝑂(2^nd ‘(𝐹‘𝑌))) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩))
11	6, 8, 10	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = ((1^st ‘(𝐹‘𝑌))𝑂(2^nd ‘(𝐹‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟨cop 4597 × cxp 5636 ∘ ccom 5642 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 1^st c1st 7924 2^nd c2nd 7925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3406 df-v 3448 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-1st 7926 df-2nd 7927

This theorem is referenced by: cnmetcoval 43544 volicoff 44356 voliooicof 44357 hoissre 44905 hoiprodcl 44908 hoicvr 44909 hoicvrrex 44917 ovn0lem 44926 ovnhoilem1 44962 ovnhoilem2 44963 hoicoto2 44966 ovnlecvr2 44971 ovncvr2 44972 ovolval2lem 45004 ovolval5lem3 45015

W3C validator