Theorem prcof22a 50014

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 50013	. . 3 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))
6	5	fveq1d 6870	. 2 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = ((𝐴 ∘ (1st ‘𝐹))‘𝑋))
7		prcof22a.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
8		eqid 2763	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	4	func1st2nd 49698	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
10	7, 8, 9	funcf1 17900	. . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
11		prcof22a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	fvco3d 6969	. 2 ⊢ (𝜑 → ((𝐴 ∘ (1st ‘𝐹))‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))
13	6, 12	eqtrd 2798	1 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ⟨cop 4589   ∘ ccom 5652  ‘cfv 6522  (class class class)co 7397  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17246   Func cfunc 17888   Nat cnat 17978   −∘_F cprcof 49995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-ixp 8881  df-func 17892  df-nat 17980  df-prcof 49996

This theorem is referenced by: (None)