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

Theorem moni 17682
Description: Property of a monomorphism. (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)
moni.z (𝜑𝑍𝐵)
moni.f (𝜑𝐹 ∈ (𝑋𝑀𝑌))
moni.g (𝜑𝐺 ∈ (𝑍𝐻𝑋))
moni.k (𝜑𝐾 ∈ (𝑍𝐻𝑋))
Assertion
Ref Expression
moni (𝜑 → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) ↔ 𝐺 = 𝐾))

Proof of Theorem moni
Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 moni.f . . . . 5 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
2 ismon.b . . . . . 6 𝐵 = (Base‘𝐶)
3 ismon.h . . . . . 6 𝐻 = (Hom ‘𝐶)
4 ismon.o . . . . . 6 · = (comp‘𝐶)
5 ismon.s . . . . . 6 𝑀 = (Mono‘𝐶)
6 ismon.c . . . . . 6 (𝜑𝐶 ∈ Cat)
7 ismon.x . . . . . 6 (𝜑𝑋𝐵)
8 ismon.y . . . . . 6 (𝜑𝑌𝐵)
92, 3, 4, 5, 6, 7, 8ismon2 17680 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))))
101, 9mpbid 231 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = )))
1110simprd 495 . . 3 (𝜑 → ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))
12 moni.z . . . 4 (𝜑𝑍𝐵)
13 moni.g . . . . . . 7 (𝜑𝐺 ∈ (𝑍𝐻𝑋))
1413adantr 480 . . . . . 6 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15 simpr 484 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1615oveq1d 7416 . . . . . 6 ((𝜑𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
1714, 16eleqtrrd 2828 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18 moni.k . . . . . . . . 9 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1918adantr 480 . . . . . . . 8 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
2019, 16eleqtrrd 2828 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
2120adantr 480 . . . . . 6 (((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22 simpllr 773 . . . . . . . . . . 11 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑧 = 𝑍)
2322opeq1d 4871 . . . . . . . . . 10 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
2423oveq1d 7416 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (⟨𝑧, 𝑋· 𝑌) = (⟨𝑍, 𝑋· 𝑌))
25 eqidd 2725 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝐹 = 𝐹)
26 simplr 766 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑔 = 𝐺)
2724, 25, 26oveq123d 7422 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺))
28 simpr 484 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → = 𝐾)
2924, 25, 28oveq123d 7422 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3027, 29eqeq12d 2740 . . . . . . 7 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → ((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) ↔ (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾)))
3126, 28eqeq12d 2740 . . . . . . 7 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝑔 = 𝐺 = 𝐾))
3230, 31imbi12d 344 . . . . . 6 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ) ↔ ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) → 𝐺 = 𝐾)))
3321, 32rspcdv 3596 . . . . 5 (((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → (∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ) → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) → 𝐺 = 𝐾)))
3417, 33rspcimdv 3594 . . . 4 ((𝜑𝑧 = 𝑍) → (∀𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ) → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) → 𝐺 = 𝐾)))
3512, 34rspcimdv 3594 . . 3 (𝜑 → (∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ) → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) → 𝐺 = 𝐾)))
3611, 35mpd 15 . 2 (𝜑 → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) → 𝐺 = 𝐾))
37 oveq2 7409 . 2 (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3836, 37impbid1 224 1 (𝜑 → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) ↔ 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3053  cop 4626  cfv 6533  (class class class)co 7401  Basecbs 17143  Hom chom 17207  compcco 17208  Catccat 17607  Monocmon 17674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-cat 17611  df-mon 17676
This theorem is referenced by:  epii  17689  monsect  17729  fthmon  17879  setcmon  18039
  Copyright terms: Public domain W3C validator