Theorem cofu1a 49591

Description: Value of the object part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)

Hypotheses

Ref	Expression
cofu1a.b	⊢ 𝐵 = (Base‘𝐶)
cofu1a.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
cofu1a.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
cofu1a.m	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)
cofu1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
cofu1a	⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))

Proof of Theorem cofu1a

Step	Hyp	Ref	Expression
1		cofu1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		cofu1a.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 5080	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 219	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofu1a.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6		df-br 5080	. . . 4 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 219	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
8		cofu1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	1, 4, 7, 8	cofu1 17849	. 2 ⊢ (𝜑 → ((1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))‘𝑋) = ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)))
10		cofu1a.m	. . . . 5 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)
11	10	fveq2d 6838	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (1st ‘⟨𝑀, 𝑁⟩))
12	4, 7	cofucl 17853	. . . . . . 7 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
13	10, 12	eqeltrrd 2841	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
14		df-br 5080	. . . . . 6 ⊢ (𝑀(𝐶 Func 𝐸)𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
15	13, 14	sylibr 235	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐸)𝑁)
16	15	func1st 49574	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
17	11, 16	eqtrd 2775	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑀)
18	17	fveq1d 6836	. 2 ⊢ (𝜑 → ((1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))‘𝑋) = (𝑀‘𝑋))
19	5	func1st 49574	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
20	2	func1st 49574	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	fveq1d 6836	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
22	19, 21	fveq12d 6841	. 2 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = (𝐾‘(𝐹‘𝑋)))
23	9, 18, 22	3eqtr3rd 2784	1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ⟨cop 4568   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  Basecbs 17177   Func cfunc 17819   ∘_func ccofu 17821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-map 8772  df-ixp 8843  df-cat 17632  df-cid 17633  df-func 17823  df-cofu 17825

This theorem is referenced by:  uptrlem1  49707  uptrlem3  49709  uptr2  49718