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Theorem opco1 8156

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 7441	. . 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 8042	. . . 4 ⊢ 1^st :V–onto→V
4		fof 6828	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 1^st :V⟶V)
6		opex 5478	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 7016	. 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 8034	. . . 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 6918	. 2 ⊢ (𝜑 → (𝐹‘(1^st ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐴))
14	2, 8, 13	3eqtrd 2781	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 Vcvv 3481 ⟨cop 4640 ∘ ccom 5697 ⟶wf 6565 –onto→wfo 6567 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5305 ax-nul 5315 ax-pr 5441 ax-un 7761

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3483 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-fo 6575 df-fv 6577 df-ov 7441 df-1st 8022

This theorem is referenced by: opco1i 8158

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