fvovco - Mathbox for Glauco Siliprandi

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Theorem fvovco 45155

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 7085	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊))
4		1st2nd2 8035	. . . 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 6890	. 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩))
7		fvco3 6988	. . 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 7416	. . 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 2779	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 1539 ∈ wcel 2107 ⟨cop 4612 × cxp 5663 ∘ ccom 5669 ⟶wf 6537 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 2^nd c2nd 7995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7416 df-1st 7996 df-2nd 7997

This theorem is referenced by: cnmetcoval 45164 volicoff 45967 voliooicof 45968 hoissre 46516 hoiprodcl 46519 hoicvr 46520 hoicvrrex 46528 ovn0lem 46537 ovnhoilem1 46573 ovnhoilem2 46574 hoicoto2 46577 ovnlecvr2 46582 ovncvr2 46583 ovolval2lem 46615 ovolval5lem3 46626

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