Theorem termcid2 49157

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 49156	. 2 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
9	1, 2, 3, 4	termcbasmo 49153	. . 3 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	fveq2d 6876	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑌))
11	8, 10	eqtrd 2769	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6527  (class class class)co 7399  Basecbs 17213  Hom chom 17267  Idccid 17662  TermCatctermc 49143

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 5246  ax-sep 5263  ax-nul 5273  ax-pr 5399

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-rmo 3357  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-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-cat 17665  df-cid 17666  df-thinc 49091  df-termc 49144

This theorem is referenced by: (None)