Theorem termcnatval 49540

Description: Value of natural transformations for a terminal category. (Contributed by Zhi Wang, 21-Oct-2025.)

Hypotheses

Ref	Expression
termcnatval.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcnatval.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
termcnatval.a	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
termcnatval.b	⊢ 𝐵 = (Base‘𝐶)
termcnatval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcnatval.r	⊢ 𝑅 = (𝐴‘𝑋)

Assertion

Ref	Expression
termcnatval	⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

Proof of Theorem termcnatval

Step	Hyp	Ref	Expression
1		termcnatval.n	. . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		termcnatval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
3	1, 2	nat1st2nd 17861	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
4		termcnatval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	1, 3, 4	natfn 17864	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐵)
6		termcnatval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ TermCat)
7		termcnatval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	6, 4, 7	termcbas2 49487	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑋})
9	8	fneq2d 6576	. . . 4 ⊢ (𝜑 → (𝐴 Fn 𝐵 ↔ 𝐴 Fn {𝑋}))
10	5, 9	mpbid 232	. . 3 ⊢ (𝜑 → 𝐴 Fn {𝑋})
11		fnsnbg 7100	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
12	7, 11	syl 17	. . 3 ⊢ (𝜑 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
13	10, 12	mpbid 232	. 2 ⊢ (𝜑 → 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩})
14		termcnatval.r	. . . 4 ⊢ 𝑅 = (𝐴‘𝑋)
15	14	opeq2i 4828	. . 3 ⊢ ⟨𝑋, 𝑅⟩ = ⟨𝑋, (𝐴‘𝑋)⟩
16	15	sneqi 4588	. 2 ⊢ {⟨𝑋, 𝑅⟩} = {⟨𝑋, (𝐴‘𝑋)⟩}
17	13, 16	eqtr4di 2782	1 ⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {csn 4577  ⟨cop 4583   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120   Nat cnat 17851  TermCatctermc 49477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-ixp 8825  df-func 17765  df-nat 17853  df-termc 49478

This theorem is referenced by:  diag2f1olem  49541