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Theorem opco2 8148

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 7434	. . 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 8034	. . . 4 ⊢ 2^nd :V–onto→V
4		fof 6821	. . . 4 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 2^nd :V⟶V)
6		opex 5475	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 7009	. 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 8026	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
13	12	fveq2d 6911	. 2 ⊢ (𝜑 → (𝐹‘(2^nd ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐵))
14	2, 8, 13	3eqtrd 2779	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⟨cop 4637 ∘ ccom 5693 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 2^nd c2nd 8012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-2nd 8014

This theorem is referenced by: dfwrecsOLD 8337 wfr2a 8373 dfrecs3 8411

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