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Theorem catlid 16391
Description: Left identity property of an 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 (𝜑𝐶 ∈ Cat)
catidcl.x (𝜑𝑋𝐵)
catlid.o · = (comp‘𝐶)
catlid.y (𝜑𝑌𝐵)
catlid.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
catlid (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)

Proof of Theorem catlid
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
2 catidcl.x . . 3 (𝜑𝑋𝐵)
3 simpl 472 . . . . . . . 8 ((∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
43ralimi 2981 . . . . . . 7 (∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
54a1i 11 . . . . . 6 (𝑔 ∈ (𝑌𝐻𝑌) → (∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
65ss2rabi 3717 . . . . 5 {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓}
7 catidcl.b . . . . . . 7 𝐵 = (Base‘𝐶)
8 catidcl.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
9 catlid.o . . . . . . 7 · = (comp‘𝐶)
10 catidcl.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
11 catidcl.i . . . . . . 7 1 = (Id‘𝐶)
12 catlid.y . . . . . . 7 (𝜑𝑌𝐵)
137, 8, 9, 10, 11, 12cidval 16385 . . . . . 6 (𝜑 → ( 1𝑌) = (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)))
147, 8, 9, 10, 12catideu 16383 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓))
15 riotacl2 6664 . . . . . . 7 (∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓) → (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
1614, 15syl 17 . . . . . 6 (𝜑 → (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
1713, 16eqeltrd 2730 . . . . 5 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
186, 17sseldi 3634 . . . 4 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓})
19 oveq1 6697 . . . . . . . 8 (𝑔 = ( 1𝑌) → (𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓))
2019eqeq1d 2653 . . . . . . 7 (𝑔 = ( 1𝑌) → ((𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
21202ralbidv 3018 . . . . . 6 (𝑔 = ( 1𝑌) → (∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
2221elrab 3396 . . . . 5 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} ↔ (( 1𝑌) ∈ (𝑌𝐻𝑌) ∧ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
2322simprbi 479 . . . 4 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
2418, 23syl 17 . . 3 (𝜑 → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
25 oveq1 6697 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
26 opeq1 4433 . . . . . . . 8 (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
2726oveq1d 6705 . . . . . . 7 (𝑥 = 𝑋 → (⟨𝑥, 𝑌· 𝑌) = (⟨𝑋, 𝑌· 𝑌))
2827oveqd 6707 . . . . . 6 (𝑥 = 𝑋 → (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓))
2928eqeq1d 2653 . . . . 5 (𝑥 = 𝑋 → ((( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
3025, 29raleqbidv 3182 . . . 4 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
3130rspcv 3336 . . 3 (𝑋𝐵 → (∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
322, 24, 31sylc 65 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓)
33 oveq2 6698 . . . 4 (𝑓 = 𝐹 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹))
34 id 22 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
3533, 34eqeq12d 2666 . . 3 (𝑓 = 𝐹 → ((( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹))
3635rspcv 3336 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹))
371, 32, 36sylc 65 1 (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  ∃!wreu 2943  {crab 2945  cop 4216  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-cat 16376  df-cid 16377
This theorem is referenced by:  oppccatid  16426  sectcan  16462  sectco  16463  sectmon  16489  monsect  16490  sectid  16493  invisoinvl  16497  subccatid  16553  fucidcl  16672  fuclid  16673  invfuc  16681  arwlid  16769  xpccatid  16875  evlfcl  16909  curf1cl  16915  curf2cl  16918  curfcl  16919  curfuncf  16925  uncfcurf  16926  hofcl  16946  yon12  16952  yon2  16953  yonedalem3b  16966  yonedainv  16968
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