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

Theorem ismon 17512
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 17511 . . . 4 (𝜑𝑀 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
7 simprl 768 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 770 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 7331 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
107oveq2d 7329 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝐻𝑥) = (𝑧𝐻𝑋))
117opeq2d 4820 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑋⟩)
1211, 8oveq12d 7331 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (⟨𝑧, 𝑥· 𝑦) = (⟨𝑧, 𝑋· 𝑌))
1312oveqd 7330 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))
1410, 13mpteq12dv 5176 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)))
1514cnveqd 5802 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)))
1615funeqd 6490 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))))
1716ralbidv 3171 . . . . 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 3419 . . . 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 7346 . . . . . 6 (𝑋𝐻𝑌) ∈ V
2221rabex 5269 . . . . 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, 23ovmpod 7463 . . 3 (𝜑 → (𝑋𝑀𝑌) = {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))})
2524eleq2d 2823 . 2 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))}))
26 oveq1 7320 . . . . . . 7 (𝑓 = 𝐹 → (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))
2726mpteq2dv 5187 . . . . . 6 (𝑓 = 𝐹 → (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))
2827cnveqd 5802 . . . . 5 (𝑓 = 𝐹(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))
2928funeqd 6490 . . . 4 (𝑓 = 𝐹 → (Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3029ralbidv 3171 . . 3 (𝑓 = 𝐹 → (∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) ↔ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3130elrab 3633 . 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, 31bitrdi 286 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 205  wa 396   = wceq 1540  wcel 2105  wral 3062  {crab 3404  Vcvv 3441  cop 4575  cmpt 5168  ccnv 5604  Fun wfun 6457  cfv 6463  (class class class)co 7313  Basecbs 16979  Hom chom 17040  compcco 17041  Catccat 17440  Monocmon 17507
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-1st 7874  df-2nd 7875  df-mon 17509
This theorem is referenced by:  ismon2  17513  monhom  17514  isepi  17519
  Copyright terms: Public domain W3C validator