fvovco - Mathbox for Glauco Siliprandi

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Theorem fvovco 42254

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	ffvelrnd 6856	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (𝑉 × 𝑊))
4		1st2nd2 7746	. . . 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 6672	. 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩))
7		fvco3 6761	. . 3 ⊢ ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌 ∈ 𝑋) → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌)))
8	1, 2, 7	syl2anc 587	. 2 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌)))
9		df-ov 7167	. . 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 2783	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 1542 ∈ wcel 2113 ⟨cop 4519 × cxp 5517 ∘ ccom 5523 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 1^st c1st 7705 2^nd c2nd 7706

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-1st 7707 df-2nd 7708

This theorem is referenced by: cnmetcoval 42264 volicoff 43062 voliooicof 43063 hoissre 43608 hoiprodcl 43611 hoicvr 43612 hoicvrrex 43620 ovn0lem 43629 ovnhoilem1 43665 ovnhoilem2 43666 hoicoto2 43669 ovnlecvr2 43674 ovncvr2 43675 ovolval2lem 43707 ovolval5lem3 43718

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