Theorem termcid2 50517

Description: The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcbasmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcid.h	⊢ 𝐻 = (Hom ‘𝐶)
termcid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
termcid.i	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
termcid2	⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

Proof of Theorem termcid2

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		termcbasmo.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		termcbasmo.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		termcid.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
6		termcid.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
7		termcid.i	. . 3 ⊢ 1 = (Id‘𝐶)
8	1, 2, 3, 4, 5, 6, 7	termcid 50516	. 2 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
9	1, 2, 3, 4	termcbasmo 50513	. . 3 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	fveq2d 6877	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑌))
11	8, 10	eqtrd 2795	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  Hom chom 17400  Idccid 17800  TermCatctermc 50502

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 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

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-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  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 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-cat 17803  df-cid 17804  df-thinc 50448  df-termc 50503

This theorem is used by: (None)