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Theorem opco2 8110

Description: Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)

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

**Assertion**
Ref	Expression
opco2	⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = (𝐹‘𝐵))

**Proof of Theorem opco2**
Step	Hyp	Ref	Expression
1		df-ov 7412	. . 3 ⊢ (𝐴(𝐹 ∘ 2^nd )𝐵) = ((𝐹 ∘ 2^nd )‘⟨𝐴, 𝐵⟩)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = ((𝐹 ∘ 2^nd )‘⟨𝐴, 𝐵⟩))
3		fo2nd 7996	. . . 4 ⊢ 2^nd :V–onto→V
4		fof 6806	. . . 4 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 2^nd :V⟶V)
6		opex 5465	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 6992	. 2 ⊢ (𝜑 → ((𝐹 ∘ 2^nd )‘⟨𝐴, 𝐵⟩) = (𝐹‘(2^nd ‘⟨𝐴, 𝐵⟩)))
9		opco1.exa	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		opco1.exb	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
11		op2ndg 7988	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
12	9, 10, 11	syl2anc 585	. . 3 ⊢ (𝜑 → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
13	12	fveq2d 6896	. 2 ⊢ (𝜑 → (𝐹‘(2^nd ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐵))
14	2, 8, 13	3eqtrd 2777	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⟨cop 4635 ∘ ccom 5681 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 2^nd c2nd 7974

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-ov 7412 df-2nd 7976

This theorem is referenced by: dfwrecsOLD 8298 wfr2a 8334 dfrecs3 8372

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