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Theorem ismon 17616
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 17615 . . . 4 (𝜑𝑀 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
7 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 7375 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
107oveq2d 7373 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝐻𝑥) = (𝑧𝐻𝑋))
117opeq2d 4837 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑋⟩)
1211, 8oveq12d 7375 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (⟨𝑧, 𝑥· 𝑦) = (⟨𝑧, 𝑋· 𝑌))
1312oveqd 7374 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))
1410, 13mpteq12dv 5196 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)))
1514cnveqd 5831 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)))
1615funeqd 6523 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔))))
1716ralbidv 3174 . . . . 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 3424 . . . 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 7390 . . . . . 6 (𝑋𝐻𝑌) ∈ V
2221rabex 5289 . . . . 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 7507 . . 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 7364 . . . . . . 7 (𝑓 = 𝐹 → (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))
2726mpteq2dv 5207 . . . . . 6 (𝑓 = 𝐹 → (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))
2827cnveqd 5831 . . . . 5 (𝑓 = 𝐹(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))
2928funeqd 6523 . . . 4 (𝑓 = 𝐹 → (Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3029ralbidv 3174 . . 3 (𝑓 = 𝐹 → (∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋· 𝑌)𝑔)) ↔ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔))))
3130elrab 3645 . 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 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cop 4592  cmpt 5188  ccnv 5632  Fun wfun 6490  cfv 6496  (class class class)co 7357  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Monocmon 17611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-mon 17613
This theorem is referenced by:  ismon2  17617  monhom  17618  isepi  17623
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