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 44596

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 7100	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊))
4		1st2nd2 8038	. . . 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 6906	. 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩))
7		fvco3 7002	. . 3 ⊢ ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌 ∈ 𝑋) → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌)))
8	1, 2, 7	syl2anc 582	. 2 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌)))
9		df-ov 7429	. . 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 2778	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 1533 ∈ wcel 2098 ⟨cop 4638 × cxp 5680 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 1^st c1st 7997 2^nd c2nd 7998

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-1st 7999 df-2nd 8000

This theorem is referenced by: cnmetcoval 44605 volicoff 45412 voliooicof 45413 hoissre 45961 hoiprodcl 45964 hoicvr 45965 hoicvrrex 45973 ovn0lem 45982 ovnhoilem1 46018 ovnhoilem2 46019 hoicoto2 46022 ovnlecvr2 46027 ovncvr2 46028 ovolval2lem 46060 ovolval5lem3 46071

W3C validator