Theorem cofu1a 49447

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 5101	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofu1a.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6		df-br 5101	. . . 4 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
8		cofu1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	1, 4, 7, 8	cofu1 17820	. 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 6846	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (1st ‘⟨𝑀, 𝑁⟩))
12	4, 7	cofucl 17824	. . . . . . 7 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
13	10, 12	eqeltrrd 2838	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
14		df-br 5101	. . . . . 6 ⊢ (𝑀(𝐶 Func 𝐸)𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
15	13, 14	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐸)𝑁)
16	15	func1st 49430	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
17	11, 16	eqtrd 2772	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑀)
18	17	fveq1d 6844	. 2 ⊢ (𝜑 → ((1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))‘𝑋) = (𝑀‘𝑋))
19	5	func1st 49430	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
20	2	func1st 49430	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	fveq1d 6844	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
22	19, 21	fveq12d 6849	. 2 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = (𝐾‘(𝐹‘𝑋)))
23	9, 18, 22	3eqtr3rd 2781	1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  Basecbs 17148   Func cfunc 17790   ∘_func ccofu 17792

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-rmo 3352  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-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ixp 8848  df-cat 17603  df-cid 17604  df-func 17794  df-cofu 17796

This theorem is referenced by:  uptrlem1  49563  uptrlem3  49565  uptr2  49574