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Theorem moni 17780
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 17778 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))))
101, 9mpbid 232 . . . 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 7446 . . . . . 6 ((𝜑𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
1714, 16eleqtrrd 2844 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18 moni.k . . . . . . . . 9 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1918adantr 480 . . . . . . . 8 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
2019, 16eleqtrrd 2844 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
2120adantr 480 . . . . . 6 (((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22 simpllr 776 . . . . . . . . . . 11 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑧 = 𝑍)
2322opeq1d 4879 . . . . . . . . . 10 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
2423oveq1d 7446 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (⟨𝑧, 𝑋· 𝑌) = (⟨𝑍, 𝑋· 𝑌))
25 eqidd 2738 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝐹 = 𝐹)
26 simplr 769 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑔 = 𝐺)
2724, 25, 26oveq123d 7452 . . . . . . . 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 7452 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3027, 29eqeq12d 2753 . . . . . . 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 2753 . . . . . . 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 3614 . . . . 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 3612 . . . 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 3612 . . 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 7439 . 2 (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3836, 37impbid1 225 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  cop 4632  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Monocmon 17772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-cat 17711  df-mon 17774
This theorem is referenced by:  epii  17787  monsect  17827  fthmon  17974  setcmon  18132
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