Theorem prcof21a 49750

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 17889	. . . . 5 ⊢ (𝐴 ∈ (𝐾𝑁𝐿) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝐿 ∈ (𝐷 Func 𝐸)))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝐿 ∈ (𝐷 Func 𝐸)))
5	4	simpld 494	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	4	simprd 495	. . 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 49748	. 2 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
10		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
11	10	coeq1d 5818	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ∘ (1st ‘𝐹)) = (𝐴 ∘ (1st ‘𝐹)))
12		fvexd 6857	. . 3 ⊢ (𝜑 → (1st ‘𝐹) ∈ V)
13	2, 12	coexd 7883	. 2 ⊢ (𝜑 → (𝐴 ∘ (1st ‘𝐹)) ∈ V)
14	9, 11, 2, 13	fvmptd 6957	1 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   ∘ ccom 5636  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942   Func cfunc 17790   Nat cnat 17880   −∘_F cprcof 49732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-ixp 8848  df-func 17794  df-nat 17882  df-prcof 49733

This theorem is referenced by:  prcof22a  49751  prcofdiag  49753  lanup  50000  ranup  50001