MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismon Structured version   Visualization version   GIF version

Theorem ismon 16374
Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
ismon.b 𝐵 = (Base‘𝐶)
ismon.h 𝐻 = (Hom ‘𝐶)
ismon.o · = (comp‘𝐶)
ismon.s 𝑀 = (Mono‘𝐶)
ismon.c (𝜑𝐶 ∈ Cat)
ismon.x (𝜑𝑋𝐵)
ismon.y (𝜑𝑌𝐵)
Assertion
Ref Expression
ismon (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))))
Distinct variable groups:   𝑧,𝑔,𝐵   𝜑,𝑔,𝑧   𝐶,𝑔,𝑧   𝑔,𝐻,𝑧   · ,𝑔,𝑧   𝑔,𝐹,𝑧   𝑔,𝑋,𝑧   𝑔,𝑌,𝑧
Allowed substitution hints:   𝑀(𝑧,𝑔)

Proof of Theorem ismon
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismon.b . . . . 5 𝐵 = (Base‘𝐶)
2 ismon.h . . . . 5 𝐻 = (Hom ‘𝐶)
3 ismon.o . . . . 5 · = (comp‘𝐶)
4 ismon.s . . . . 5 𝑀 = (Mono‘𝐶)
5 ismon.c . . . . 5 (𝜑𝐶 ∈ Cat)
61, 2, 3, 4, 5monfval 16373 . . . 4 (𝜑𝑀 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
7 simprl 793 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 795 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 6653 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
107oveq2d 6651 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝐻𝑥) = (𝑧𝐻𝑋))
117opeq2d 4400 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑋⟩)
1211, 8oveq12d 6653 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (⟨𝑧, 𝑥· 𝑦) = (⟨𝑧, 𝑋· 𝑌))
1312oveqd 6652 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))
1410, 13mpteq12dv 4724 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)))
1514cnveqd 5287 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)))
1615funeqd 5898 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))))
1716ralbidv 2983 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) ↔ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))))
189, 17rabeqbidv 3190 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))} = {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))})
19 ismon.x . . . 4 (𝜑𝑋𝐵)
20 ismon.y . . . 4 (𝜑𝑌𝐵)
21 ovex 6663 . . . . . 6 (𝑋𝐻𝑌) ∈ V
2221rabex 4804 . . . . 5 {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))} ∈ V
2322a1i 11 . . . 4 (𝜑 → {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))} ∈ V)
246, 18, 19, 20, 23ovmpt2d 6773 . . 3 (𝜑 → (𝑋𝑀𝑌) = {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))})
2524eleq2d 2685 . 2 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))}))
26 oveq1 6642 . . . . . . 7 (𝑓 = 𝐹 → (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))
2726mpteq2dv 4736 . . . . . 6 (𝑓 = 𝐹 → (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))
2827cnveqd 5287 . . . . 5 (𝑓 = 𝐹(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))
2928funeqd 5898 . . . 4 (𝑓 = 𝐹 → (Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3029ralbidv 2983 . . 3 (𝑓 = 𝐹 → (∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) ↔ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3130elrab 3357 . 2 (𝐹 ∈ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))} ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3225, 31syl6bb 276 1 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  cop 4174  cmpt 4720  ccnv 5103  Fun wfun 5870  cfv 5876  (class class class)co 6635  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Monocmon 16369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-mon 16371
This theorem is referenced by:  ismon2  16375  monhom  16376  isepi  16381
  Copyright terms: Public domain W3C validator