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Theorem cofurid 16322
Description: The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofulid.g (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofurid.1 𝐼 = (idfunc𝐶)
Assertion
Ref Expression
cofurid (𝜑 → (𝐹func 𝐼) = 𝐹)

Proof of Theorem cofurid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofurid.1 . . . . . 6 𝐼 = (idfunc𝐶)
2 eqid 2609 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
3 cofulid.g . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
4 funcrcl 16294 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
65simpld 473 . . . . . 6 (𝜑𝐶 ∈ Cat)
71, 2, 6idfu1st 16310 . . . . 5 (𝜑 → (1st𝐼) = ( I ↾ (Base‘𝐶)))
87coeq2d 5193 . . . 4 (𝜑 → ((1st𝐹) ∘ (1st𝐼)) = ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))))
9 eqid 2609 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
10 relfunc 16293 . . . . . . 7 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7085 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 3, 11sylancr 693 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
132, 9, 12funcf1 16297 . . . . 5 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
14 fcoi1 5975 . . . . 5 ((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) → ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st𝐹))
1513, 14syl 17 . . . 4 (𝜑 → ((1st𝐹) ∘ ( I ↾ (Base‘𝐶))) = (1st𝐹))
168, 15eqtrd 2643 . . 3 (𝜑 → ((1st𝐹) ∘ (1st𝐼)) = (1st𝐹))
1773ad2ant1 1074 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐼) = ( I ↾ (Base‘𝐶)))
1817fveq1d 6089 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑥) = (( I ↾ (Base‘𝐶))‘𝑥))
19 fvresi 6321 . . . . . . . . . 10 (𝑥 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
20193ad2ant2 1075 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑥) = 𝑥)
2118, 20eqtrd 2643 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑥) = 𝑥)
2217fveq1d 6089 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑦) = (( I ↾ (Base‘𝐶))‘𝑦))
23 fvresi 6321 . . . . . . . . . 10 (𝑦 ∈ (Base‘𝐶) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
24233ad2ant3 1076 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (( I ↾ (Base‘𝐶))‘𝑦) = 𝑦)
2522, 24eqtrd 2643 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐼)‘𝑦) = 𝑦)
2621, 25oveq12d 6544 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) = (𝑥(2nd𝐹)𝑦))
2763ad2ant1 1074 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
28 eqid 2609 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
29 simp2 1054 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
30 simp3 1055 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
311, 2, 27, 28, 29, 30idfu2nd 16308 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐶)𝑦)))
3226, 31coeq12d 5195 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)) = ((𝑥(2nd𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))))
33 eqid 2609 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
34123ad2ant1 1074 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
352, 28, 33, 34, 29, 30funcf2 16299 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
36 fcoi1 5975 . . . . . . 7 ((𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) → ((𝑥(2nd𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))) = (𝑥(2nd𝐹)𝑦))
3735, 36syl 17 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐹)𝑦) ∘ ( I ↾ (𝑥(Hom ‘𝐶)𝑦))) = (𝑥(2nd𝐹)𝑦))
3832, 37eqtrd 2643 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)) = (𝑥(2nd𝐹)𝑦))
3938mpt2eq3dva 6594 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
402, 12funcfn2 16300 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
41 fnov 6643 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4240, 41sylib 206 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
4339, 42eqtr4d 2646 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦))) = (2nd𝐹))
4416, 43opeq12d 4342 . 2 (𝜑 → ⟨((1st𝐹) ∘ (1st𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
451idfucl 16312 . . . 4 (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
466, 45syl 17 . . 3 (𝜑𝐼 ∈ (𝐶 Func 𝐶))
472, 46, 3cofuval 16313 . 2 (𝜑 → (𝐹func 𝐼) = ⟨((1st𝐹) ∘ (1st𝐼)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐼)‘𝑥)(2nd𝐹)((1st𝐼)‘𝑦)) ∘ (𝑥(2nd𝐼)𝑦)))⟩)
48 1st2nd 7082 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
4910, 3, 48sylancr 693 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5044, 47, 493eqtr4d 2653 1 (𝜑 → (𝐹func 𝐼) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  cop 4130   class class class wbr 4577   I cid 4937   × cxp 5025  cres 5029  ccom 5031  Rel wrel 5032   Fn wfn 5784  wf 5785  cfv 5789  (class class class)co 6526  cmpt2 6528  1st c1st 7034  2nd c2nd 7035  Basecbs 15643  Hom chom 15727  Catccat 16096   Func cfunc 16285  idfunccidfu 16286  func ccofu 16287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16100  df-cid 16101  df-func 16289  df-idfu 16290  df-cofu 16291
This theorem is referenced by:  catccatid  16523
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