Theorem catidcl 17849

Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
catidcl.b	⊢ 𝐵 = (Base‘𝐶)
catidcl.h	⊢ 𝐻 = (Hom ‘𝐶)
catidcl.i	⊢ 1 = (Id‘𝐶)
catidcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
catidcl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
catidcl	⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))

Proof of Theorem catidcl

Dummy variables 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catidcl.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		catidcl.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2761	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		catidcl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		catidcl.i	. . 3 ⊢ 1 = (Id‘𝐶)
6		catidcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	cidval 17844	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
8	1, 2, 3, 4, 6	catideu 17842	. . 3 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
9		riotacl 7392	. . 3 ⊢ (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋))
10	8, 9	syl 18	. 2 ⊢ (𝜑 → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩(comp‘𝐶)𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ (𝑋𝐻𝑋))
11	7, 10	eqeltrd 2861	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364  ⟨cop 4590  ‘cfv 6537  ℩crio 7374  (class class class)co 7418  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831  Idccid 17832

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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-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 5546  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-cat 17835  df-cid 17836

This theorem is used by:  oppccatid  17886  monsect  17951  sectid  17954  catsubcat  18007  fullsubc  18018  idfucl  18049  cofucl  18056  fthsect  18095  fucidcl  18136  initoid  18169  termoid  18170  idahom  18228  catcisolem  18278  xpccatid  18355  1stfcl  18364  2ndfcl  18365  prfcl  18370  evlfcl  18389  curf1cl  18395  curf2cl  18398  curfcl  18399  curfuncf  18405  uncfcurf  18406  diag12  18411  diag2  18412  curf2ndf  18414  hofcl  18426  yon12  18432  yon2  18433  yonedalem3a  18441  yonedalem3b  18446  yonedainv  18448  bj-endmnd  38219  catprs  50088  endmndlem  50092  idmon  50097  idepi  50098  discsubc  50141  imaid  50231  upciclem3  50245  swapfid  50356  tposcurf12  50375  tposcurf2  50377  fucoid  50425  thincid  50509  functhinclem4  50524  termchomn0  50561  idfudiag1  50602  termcarweu  50605  arweutermc  50607