Theorem termccofix 50762

Description: Post-composition by an object-fixing endofunctor leaves a functor with terminal source literally unchanged. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
termcfunc1eq.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcfunc1eq.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
termccofix.s	⊢ (𝜑 → 𝑆 ∈ (𝐷 Func 𝐷))
termccofix.i	⊢ (𝜑 → (1st ‘𝑆) = ( I ↾ 𝐴))

Assertion

Ref	Expression
termccofix	⊢ (𝜑 → (𝑆 ∘func 𝐹) = 𝐹)

Proof of Theorem termccofix

Step	Hyp	Ref	Expression
1		termcfunc1eq.c	. 2 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcfunc1eq.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		termccofix.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐷 Func 𝐷))
4	2, 3	cofucl 18065	. 2 ⊢ (𝜑 → (𝑆 ∘func 𝐹) ∈ (𝐶 Func 𝐷))
5		eqid 2760	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 5	termco 50707	. 2 ⊢ (𝜑 → ∪ (Base‘𝐶) ∈ (Base‘𝐶))
7	5, 2, 3, 6	cofu1 18061	. . 3 ⊢ (𝜑 → ((1st ‘(𝑆 ∘func 𝐹))‘∪ (Base‘𝐶)) = ((1st ‘𝑆)‘((1st ‘𝐹)‘∪ (Base‘𝐶))))
8		termccofix.i	. . . 4 ⊢ (𝜑 → (1st ‘𝑆) = ( I ↾ 𝐴))
9	8	fveq1d 6887	. . 3 ⊢ (𝜑 → ((1st ‘𝑆)‘((1st ‘𝐹)‘∪ (Base‘𝐶))) = (( I ↾ 𝐴)‘((1st ‘𝐹)‘∪ (Base‘𝐶))))
10		eqid 2760	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
11	2	func1st2nd 50281	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	5, 10, 11	funcf1 18043	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
13	12, 6	ffvelcdmd 7085	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐹)‘∪ (Base‘𝐶)) ∈ (Base‘𝐷))
14	8	dmeqd 5886	. . . . . . 7 ⊢ (𝜑 → dom (1st ‘𝑆) = dom ( I ↾ 𝐴))
15		dmresi 6043	. . . . . . 7 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	14, 15	eqtrdi 2811	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝑆) = 𝐴)
17	3	func1st2nd 50281	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑆)(𝐷 Func 𝐷)(2nd ‘𝑆))
18	10, 10, 17	funcf1 18043	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑆):(Base‘𝐷)⟶(Base‘𝐷))
19	18	fdmd 6720	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝑆) = (Base‘𝐷))
20	16, 19	eqtr3d 2797	. . . . 5 ⊢ (𝜑 → 𝐴 = (Base‘𝐷))
21	13, 20	eleqtrrd 2863	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹)‘∪ (Base‘𝐶)) ∈ 𝐴)
22		fvresi 7178	. . . 4 ⊢ (((1st ‘𝐹)‘∪ (Base‘𝐶)) ∈ 𝐴 → (( I ↾ 𝐴)‘((1st ‘𝐹)‘∪ (Base‘𝐶))) = ((1st ‘𝐹)‘∪ (Base‘𝐶)))
23	21, 22	syl 18	. . 3 ⊢ (𝜑 → (( I ↾ 𝐴)‘((1st ‘𝐹)‘∪ (Base‘𝐶))) = ((1st ‘𝐹)‘∪ (Base‘𝐶)))
24	7, 9, 23	3eqtrd 2799	. 2 ⊢ (𝜑 → ((1st ‘(𝑆 ∘func 𝐹))‘∪ (Base‘𝐶)) = ((1st ‘𝐹)‘∪ (Base‘𝐶)))
25	1, 4, 5, 2, 6, 24	termcfunc1eq 50761	1 ⊢ (𝜑 → (𝑆 ∘func 𝐹) = 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∪ cuni 4866   I cid 5544  dom cdm 5650   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17389   Func cfunc 18031   ∘_func ccofu 18033  TermCatctermc 50698

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-rmo 3365  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-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8851  df-ixp 8928  df-cat 17844  df-cid 17845  df-func 18035  df-cofu 18037  df-thinc 50644  df-termc 50699

This theorem is used by:  termcpostfix  50763