Theorem termcnatval 49281

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 17954	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
4		termcnatval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	1, 3, 4	natfn 17957	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐵)
6		termcnatval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ TermCat)
7		termcnatval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	6, 4, 7	termcbas2 49228	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑋})
9	8	fneq2d 6629	. . . 4 ⊢ (𝜑 → (𝐴 Fn 𝐵 ↔ 𝐴 Fn {𝑋}))
10	5, 9	mpbid 232	. . 3 ⊢ (𝜑 → 𝐴 Fn {𝑋})
11		fnsnbg 7153	. . . 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 4851	. . 3 ⊢ ⟨𝑋, 𝑅⟩ = ⟨𝑋, (𝐴‘𝑋)⟩
16	15	sneqi 4610	. 2 ⊢ {⟨𝑋, 𝑅⟩} = {⟨𝑋, (𝐴‘𝑋)⟩}
17	13, 16	eqtr4di 2787	1 ⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  {csn 4599  ⟨cop 4605   Fn wfn 6523  ‘cfv 6528  (class class class)co 7400  1^st c1st 7981  2^nd c2nd 7982  Basecbs 17215   Nat cnat 17944  TermCatctermc 49219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-ixp 8907  df-func 17858  df-nat 17946  df-termc 49220

This theorem is referenced by:  diag2f1olem  49282