Theorem prcof21a 50499

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)
prcof21a.f	⊢ (𝜑 → 𝐹 ∈ 𝑈)

Assertion

Ref	Expression
prcof21a	⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

Proof of Theorem prcof21a

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prcof21a.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐸)
2		prcof21a.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))
3	1	natrcl 18128	. . . . 5 ⊢ (𝐴 ∈ (𝐾𝑁𝐿) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝐿 ∈ (𝐷 Func 𝐸)))
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝐿 ∈ (𝐷 Func 𝐸)))
5	4	simpld 500	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	4	simprd 501	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
7		prcof21a.p	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
8		prcof21a.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
9	1, 5, 6, 7, 8	prcof2a 50497	. 2 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
10		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
11	10	coeq1d 5838	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ∘ (1st ‘𝐹)) = (𝐴 ∘ (1st ‘𝐹)))
12		fvexd 6900	. . 3 ⊢ (𝜑 → (1st ‘𝐹) ∈ V)
13	2, 12	coexd 7943	. 2 ⊢ (𝜑 → (𝐴 ∘ (1st ‘𝐹)) ∈ V)
14	9, 11, 2, 13	fvmptd 7001	1 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4589   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000   Func cfunc 18029   Nat cnat 18119   −∘_F cprcof 50481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-ixp 8926  df-func 18033  df-nat 18121  df-prcof 50482

This theorem is used by:  prcof22a  50500  prcofdiag  50502  lanup  50749  ranup  50750