Theorem termcid2 49456

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 49455	. 2 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
9	1, 2, 3, 4	termcbasmo 49452	. . 3 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	fveq2d 6864	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑌))
11	8, 10	eqtrd 2765	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  Hom chom 17237  Idccid 17632  TermCatctermc 49441

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-cat 17635  df-cid 17636  df-thinc 49387  df-termc 49442

This theorem is referenced by: (None)