fvovco - Mathbox for Glauco Siliprandi

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Theorem fvovco 45581

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 7984	. . . 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 6848	. 2 ⊢ (𝜑 → (𝑂‘(𝐹‘𝑌)) = (𝑂‘⟨(1^st ‘(𝐹‘𝑌)), (2^nd ‘(𝐹‘𝑌))⟩))
7		fvco3 6943	. . 3 ⊢ ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌 ∈ 𝑋) → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌)))
8	1, 2, 7	syl2anc 585	. 2 ⊢ (𝜑 → ((𝑂 ∘ 𝐹)‘𝑌) = (𝑂‘(𝐹‘𝑌)))
9		df-ov 7373	. . 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 2782	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 2114 ⟨cop 4588 × cxp 5632 ∘ ccom 5638 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 1^st c1st 7943 2^nd c2nd 7944

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pr 5381 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-fv 6510 df-ov 7373 df-1st 7945 df-2nd 7946

This theorem is referenced by: cnmetcoval 45589 volicoff 46382 voliooicof 46383 hoissre 46931 hoiprodcl 46934 hoicvr 46935 hoicvrrex 46943 ovn0lem 46952 ovnhoilem1 46988 ovnhoilem2 46989 hoicoto2 46992 ovnlecvr2 46997 ovncvr2 46998 ovolval2lem 47030 ovolval5lem3 47041

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