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Theorem opco2 8054

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 7349	. . 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 7942	. . . 4 ⊢ 2^nd :V–onto→V
4		fof 6735	. . . 4 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 2^nd :V⟶V)
6		opex 5402	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 6922	. 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 7934	. . . 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 6826	. 2 ⊢ (𝜑 → (𝐹‘(2^nd ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐵))
14	2, 8, 13	3eqtrd 2770	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ⟨cop 4579 ∘ ccom 5618 ⟶wf 6477 –onto→wfo 6479 ‘cfv 6481 (class class class)co 7346 2^nd c2nd 7920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-ov 7349 df-2nd 7922

This theorem is referenced by: wfr2a 8255 dfrecs3 8292

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