Theorem termcnatval 50559

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 18076	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
4		termcnatval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	1, 3, 4	natfn 18079	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐵)
6		termcnatval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ TermCat)
7		termcnatval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	6, 4, 7	termcbas2 50506	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑋})
9	8	fneq2d 6622	. . . 4 ⊢ (𝜑 → (𝐴 Fn 𝐵 ↔ 𝐴 Fn {𝑋}))
10	5, 9	mpbid 235	. . 3 ⊢ (𝜑 → 𝐴 Fn {𝑋})
11		fnsnbg 7158	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
12	7, 11	syl 18	. . 3 ⊢ (𝜑 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
13	10, 12	mpbid 235	. 2 ⊢ (𝜑 → 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩})
14		termcnatval.r	. . . 4 ⊢ 𝑅 = (𝐴‘𝑋)
15	14	opeq2i 4837	. . 3 ⊢ ⟨𝑋, 𝑅⟩ = ⟨𝑋, (𝐴‘𝑋)⟩
16	15	sneqi 4595	. 2 ⊢ {⟨𝑋, 𝑅⟩} = {⟨𝑋, (𝐴‘𝑋)⟩}
17	13, 16	eqtr4di 2813	1 ⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {csn 4584  ⟨cop 4590   Fn wfn 6523  ‘cfv 6528  (class class class)co 7409  1^st c1st 7983  2^nd c2nd 7984  Basecbs 17334   Nat cnat 18066  TermCatctermc 50496

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7985  df-2nd 7986  df-ixp 8905  df-func 17980  df-nat 18068  df-termc 50497

This theorem is used by:  diag2f1olem  50560