opco2 - Metamath Proof Explorer

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Theorem opco2 8128

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 7413	. . 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 8014	. . . 4 ⊢ 2^nd :V–onto→V
4		fof 6795	. . . 4 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → 2^nd :V⟶V)
6		opex 5444	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
8	5, 7	fvco3d 6984	. 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 8006	. . . 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 6885	. 2 ⊢ (𝜑 → (𝐹‘(2^nd ‘⟨𝐴, 𝐵⟩)) = (𝐹‘𝐵))
14	2, 8, 13	3eqtrd 2775	1 ⊢ (𝜑 → (𝐴(𝐹 ∘ 2^nd )𝐵) = (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ⟨cop 4612 ∘ ccom 5663 ⟶wf 6532 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7410 2^nd c2nd 7992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-ov 7413 df-2nd 7994

This theorem is referenced by: dfwrecsOLD 8317 wfr2a 8353 dfrecs3 8391

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