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Theorem opco1 8063

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 7360	. . 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 7952	. . . 4 ⊢ 1^st :V–onto→V
4		fof 6740	. . . 4 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 1^st :V⟶V)
6		opex 5404	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 6929	. 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 7944	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
12	9, 10, 11	syl2anc 590	. . 3 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
13	12	fveq2d 6832	. 2 ⊢ (𝜑 → (𝐹‘(1^st ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐴))
14	2, 8, 13	3eqtrd 2778	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 1^st )𝐵) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ⟨cop 4562 ∘ ccom 5623 ⟶wf 6482 –onto→wfo 6484 ‘cfv 6486 (class class class)co 7357 1^st c1st 7930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pr 5363 ax-un 7679

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4263 df-if 4456 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fo 6492 df-fv 6494 df-ov 7360 df-1st 7932

This theorem is referenced by: opco1i 8065

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