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Theorem moni 16219
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 16217 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))))
101, 9mpbid 221 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = )))
1110simprd 478 . . 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 476 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1615oveq1d 6564 . . . . . 6 ((𝜑𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
1714, 16eleqtrrd 2691 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18 moni.k . . . . . . . . 9 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1918adantr 480 . . . . . . . 8 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
2019, 16eleqtrrd 2691 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
2120adantr 480 . . . . . 6 (((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22 simpllr 795 . . . . . . . . . . 11 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑧 = 𝑍)
2322opeq1d 4346 . . . . . . . . . 10 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
2423oveq1d 6564 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (⟨𝑧, 𝑋· 𝑌) = (⟨𝑍, 𝑋· 𝑌))
25 eqidd 2611 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝐹 = 𝐹)
26 simplr 788 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑔 = 𝐺)
2724, 25, 26oveq123d 6570 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺))
28 simpr 476 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → = 𝐾)
2924, 25, 28oveq123d 6570 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3027, 29eqeq12d 2625 . . . . . . 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 2625 . . . . . . 7 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝑔 = 𝐺 = 𝐾))
3230, 31imbi12d 333 . . . . . 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 3285 . . . . 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 3283 . . . 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 3283 . . 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 6557 . 2 (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3836, 37impbid1 214 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cop 4131  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Monocmon 16211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-mon 16213
This theorem is referenced by:  epii  16226  monsect  16266  fthmon  16410  setcmon  16560
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