Theorem prcof22a 49371

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 49370	. . 3 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))
6	5	fveq1d 6862	. 2 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = ((𝐴 ∘ (1st ‘𝐹))‘𝑋))
7		prcof22a.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
8		eqid 2730	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	4	func1st2nd 49055	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
10	7, 8, 9	funcf1 17834	. . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷))
11		prcof22a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	10, 11	fvco3d 6963	. 2 ⊢ (𝜑 → ((𝐴 ∘ (1st ‘𝐹))‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))
13	6, 12	eqtrd 2765	1 ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4597   ∘ ccom 5644  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  Basecbs 17185   Func cfunc 17822   Nat cnat 17912   −∘_F cprcof 49352

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-ixp 8873  df-func 17826  df-nat 17914  df-prcof 49353

This theorem is referenced by: (None)