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Theorem opco2 8165

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 7451	. . 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 8051	. . . 4 ⊢ 2^nd :V–onto→V
4		fof 6834	. . . 4 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 2^nd :V⟶V)
6		opex 5484	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 7022	. 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 8043	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
12	9, 10, 11	syl2anc 583	. . 3 ⊢ (𝜑 → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
13	12	fveq2d 6924	. 2 ⊢ (𝜑 → (𝐹‘(2^nd ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐵))
14	2, 8, 13	3eqtrd 2784	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ⟨cop 4654 ∘ ccom 5704 ⟶wf 6569 –onto→wfo 6571 ‘cfv 6573 (class class class)co 7448 2^nd c2nd 8029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579 df-fv 6581 df-ov 7451 df-2nd 8031

This theorem is referenced by: dfwrecsOLD 8354 wfr2a 8390 dfrecs3 8428

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