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Theorem opco1 7932

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 7255	. . 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 7821	. . . 4 ⊢ 1^st :V–onto→V
4		fof 6669	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 1^st :V⟶V)
6		opex 5372	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 6847	. 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 7813	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
12	9, 10, 11	syl2anc 587	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
13	12	fveq2d 6757	. 2 ⊢ (𝜑 → (𝐹‘(1^st ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐴))
14	2, 8, 13	3eqtrd 2783	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3423 ⟨cop 4564 ∘ ccom 5583 ⟶wf 6411 –onto→wfo 6413 ‘cfv 6415 (class class class)co 7252 1^st c1st 7799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5216 ax-nul 5223 ax-pr 5346 ax-un 7563

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5479 df-xp 5585 df-rel 5586 df-cnv 5587 df-co 5588 df-dm 5589 df-rn 5590 df-res 5591 df-ima 5592 df-iota 6373 df-fun 6417 df-fn 6418 df-f 6419 df-fo 6421 df-fv 6423 df-ov 7255 df-1st 7801

This theorem is referenced by: opco1i 7934

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