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Theorem opco1 8137

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 7427	. . 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 8023	. . . 4 ⊢ 1^st :V–onto→V
4		fof 6815	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 1^st :V⟶V)
6		opex 5470	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 7002	. 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 8015	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
12	9, 10, 11	syl2anc 582	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
13	12	fveq2d 6905	. 2 ⊢ (𝜑 → (𝐹‘(1^st ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐴))
14	2, 8, 13	3eqtrd 2770	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⟨cop 4639 ∘ ccom 5686 ⟶wf 6550 –onto→wfo 6552 ‘cfv 6554 (class class class)co 7424 1^st c1st 8001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fo 6560 df-fv 6562 df-ov 7427 df-1st 8003

This theorem is referenced by: opco1i 8139

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