Theorem prcof22a 49579

Description: The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)

Hypotheses

Ref	Expression
prcof21a.n	⊢ 𝑁 = (𝐷 Nat 𝐸)
prcof21a.a	⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))
prcof21a.p	⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
prcof22a.b	⊢ 𝐵 = (Base‘𝐶)
prcof22a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
prcof22a.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Assertion

Ref	Expression
prcof22a	⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))

Proof of Theorem prcof22a

Step	Hyp	Ref	Expression
1		prcof21a.n	. . . 4 ⊢ 𝑁 = (𝐷 Nat 𝐸)
2		prcof21a.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))
3		prcof21a.p	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
4		prcof22a.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	1, 2, 3, 4	prcof21a 49578	. . 3 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))
6	5	fveq1d 6834	. 2 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = ((𝐴 ∘ (1st ‘𝐹))‘𝑋))
7		prcof22a.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
8		eqid 2734	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	4	func1st2nd 49263	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
10	7, 8, 9	funcf1 17788	. . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
11		prcof22a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	fvco3d 6932	. 2 ⊢ (𝜑 → ((𝐴 ∘ (1st ‘𝐹))‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))
13	6, 12	eqtrd 2769	1 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4584   ∘ ccom 5626  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17134   Func cfunc 17776   Nat cnat 17866   −∘_F cprcof 49560

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-ixp 8834  df-func 17780  df-nat 17868  df-prcof 49561

This theorem is referenced by: (None)