Theorem cofu1stfix 50303

Description: Post-composition by an object-fixing functor preserves the entire object function, including its exact domain. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
cofu1stfix.a	⊢ 𝐴 = (Base‘𝐸)
cofu1stfix.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
cofu1stfix.k	⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐸))
cofu1stfix.i	⊢ (𝜑 → (1st ‘𝐾) = ( I ↾ 𝐴))

Assertion

Ref	Expression
cofu1stfix	⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐹))

Proof of Theorem cofu1stfix

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		cofu1stfix.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
3		cofu1stfix.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐸))
4	1, 2, 3	cofu1st 18060	. 2 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = ((1st ‘𝐾) ∘ (1st ‘𝐹)))
5		cofu1stfix.i	. . 3 ⊢ (𝜑 → (1st ‘𝐾) = ( I ↾ 𝐴))
6	5	coeq1d 5838	. 2 ⊢ (𝜑 → ((1st ‘𝐾) ∘ (1st ‘𝐹)) = (( I ↾ 𝐴) ∘ (1st ‘𝐹)))
7		cofu1stfix.a	. . . 4 ⊢ 𝐴 = (Base‘𝐸)
8	2	func1st2nd 50281	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
9	1, 7, 8	funcf1 18043	. . 3 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐷)⟶𝐴)
10		fcoi2 6757	. . 3 ⊢ ((1st ‘𝐹):(Base‘𝐷)⟶𝐴 → (( I ↾ 𝐴) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
11	9, 10	syl 18	. 2 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
12	4, 6, 11	3eqtrd 2799	1 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   I cid 5544   ↾ cres 5652   ∘ ccom 5654  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17389   Func cfunc 18031   ∘_func ccofu 18033

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-map 8851  df-ixp 8928  df-func 18035  df-cofu 18037

This theorem is used by:  fucoaltfunc  50582