Theorem prcof22a 50198

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 50197	. . 3 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))
6	5	fveq1d 6883	. 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 49882	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
10	7, 8, 9	funcf1 17927	. . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
11		prcof22a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	fvco3d 6982	. 2 ⊢ (𝜑 → ((𝐴 ∘ (1st ‘𝐹))‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))
13	6, 12	eqtrd 2798	1 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17273   Func cfunc 17915   Nat cnat 18005   −∘_F cprcof 50179

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-func 17919  df-nat 18007  df-prcof 50180

This theorem is used by: (None)