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Theorem catlid 16933
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 oveq2 7141 . . 3 (𝑓 = 𝐹 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹))
2 id 22 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
31, 2eqeq12d 2836 . 2 (𝑓 = 𝐹 → ((( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹))
4 oveq1 7140 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌))
5 opeq1 4779 . . . . . . 7 (𝑥 = 𝑋 → ⟨𝑥, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
65oveq1d 7148 . . . . . 6 (𝑥 = 𝑋 → (⟨𝑥, 𝑌· 𝑌) = (⟨𝑋, 𝑌· 𝑌))
76oveqd 7150 . . . . 5 (𝑥 = 𝑋 → (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓))
87eqeq1d 2822 . . . 4 (𝑥 = 𝑋 → ((( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
94, 8raleqbidv 3388 . . 3 (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓))
10 simpl 485 . . . . . . . 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)
1110ralimi 3147 . . . . . . 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)
1211a1i 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))
1312ss2rabi 4032 . . . . 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}
14 catidcl.b . . . . . . 7 𝐵 = (Base‘𝐶)
15 catidcl.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
16 catlid.o . . . . . . 7 · = (comp‘𝐶)
17 catidcl.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
18 catidcl.i . . . . . . 7 1 = (Id‘𝐶)
19 catlid.y . . . . . . 7 (𝜑𝑌𝐵)
2014, 15, 16, 17, 18, 19cidval 16927 . . . . . 6 (𝜑 → ( 1𝑌) = (𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)))
2114, 15, 16, 17, 19catideu 16925 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓))
22 riotacl2 7107 . . . . . . 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)})
2321, 22syl 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)})
2420, 23eqeltrd 2911 . . . . 5 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(⟨𝑌, 𝑌· 𝑥)𝑔) = 𝑓)})
2513, 24sseldi 3944 . . . 4 (𝜑 → ( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓})
26 oveq1 7140 . . . . . . . 8 (𝑔 = ( 1𝑌) → (𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓))
2726eqeq1d 2822 . . . . . . 7 (𝑔 = ( 1𝑌) → ((𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ (( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
28272ralbidv 3186 . . . . . 6 (𝑔 = ( 1𝑌) → (∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓 ↔ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
2928elrab 3660 . . . . 5 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} ↔ (( 1𝑌) ∈ (𝑌𝐻𝑌) ∧ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓))
3029simprbi 499 . . . 4 (( 1𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(𝑔(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓} → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
3125, 30syl 17 . . 3 (𝜑 → ∀𝑥𝐵𝑓 ∈ (𝑥𝐻𝑌)(( 1𝑌)(⟨𝑥, 𝑌· 𝑌)𝑓) = 𝑓)
32 catidcl.x . . 3 (𝜑𝑋𝐵)
339, 31, 32rspcdva 3604 . 2 (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝑓) = 𝑓)
34 catlid.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
353, 33, 34rspcdva 3604 1 (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3125  ∃!wreu 3127  {crab 3129  cop 4549  cfv 6331  crio 7090  (class class class)co 7133  Basecbs 16462  Hom chom 16555  compcco 16556  Catccat 16914  Idccid 16915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-cat 16918  df-cid 16919
This theorem is referenced by:  oppccatid  16968  sectcan  17004  sectco  17005  sectmon  17031  monsect  17032  sectid  17035  invisoinvl  17039  subccatid  17095  fucidcl  17214  fuclid  17215  invfuc  17223  arwlid  17311  xpccatid  17417  evlfcl  17451  curf1cl  17457  curf2cl  17460  curfcl  17461  curfuncf  17467  uncfcurf  17468  hofcl  17488  yon12  17494  yon2  17495  yonedalem3b  17508  yonedainv  17510  bj-endmnd  34616
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