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Theorem opco1 7953

Description: Value of an operation precomposed with the projection on the first component. (Contributed by Mario Carneiro, 28-May-2014.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)

**Hypotheses**
Ref	Expression
opco1.exa	⊢ (𝜑 → 𝐴 ∈ 𝑉)
opco1.exb	⊢ (𝜑 → 𝐵 ∈ 𝑊)

**Assertion**
Ref	Expression
opco1	⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = (𝐹‘𝐴))

**Proof of Theorem opco1**
Step	Hyp	Ref	Expression
1		df-ov 7272	. . 3 ⊢ (𝐴(𝐹 ∘ 1^st )𝐵) = ((𝐹 ∘ 1^st )‘⟨𝐴, 𝐵⟩)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = ((𝐹 ∘ 1^st )‘⟨𝐴, 𝐵⟩))
3		fo1st 7842	. . . 4 ⊢ 1^st :V–onto→V
4		fof 6682	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 1^st :V⟶V)
6		opex 5379	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 6862	. 2 ⊢ (𝜑 → ((𝐹 ∘ 1^st )‘⟨𝐴, 𝐵⟩) = (𝐹‘(1^st ‘⟨𝐴, 𝐵⟩)))
9		opco1.exa	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		opco1.exb	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
11		op1stg 7834	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
13	12	fveq2d 6772	. 2 ⊢ (𝜑 → (𝐹‘(1^st ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐴))
14	2, 8, 13	3eqtrd 2782	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3431 ⟨cop 4569 ∘ ccom 5590 ⟶wf 6424 –onto→wfo 6426 ‘cfv 6428 (class class class)co 7269 1^st c1st 7820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7580

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-fo 6434 df-fv 6436 df-ov 7272 df-1st 7822

This theorem is referenced by: opco1i 7955

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