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Theorem moni 16996
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 16994 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))))
101, 9mpbid 233 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = )))
1110simprd 496 . . 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 481 . . . . . 6 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15 simpr 485 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1615oveq1d 7163 . . . . . 6 ((𝜑𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
1714, 16eleqtrrd 2921 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18 moni.k . . . . . . . . 9 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1918adantr 481 . . . . . . . 8 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
2019, 16eleqtrrd 2921 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
2120adantr 481 . . . . . 6 (((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22 simpllr 772 . . . . . . . . . . 11 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑧 = 𝑍)
2322opeq1d 4808 . . . . . . . . . 10 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
2423oveq1d 7163 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (⟨𝑧, 𝑋· 𝑌) = (⟨𝑍, 𝑋· 𝑌))
25 eqidd 2827 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝐹 = 𝐹)
26 simplr 765 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑔 = 𝐺)
2724, 25, 26oveq123d 7169 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺))
28 simpr 485 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → = 𝐾)
2924, 25, 28oveq123d 7169 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3027, 29eqeq12d 2842 . . . . . . 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 2842 . . . . . . 7 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝑔 = 𝐺 = 𝐾))
3230, 31imbi12d 346 . . . . . 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 3619 . . . . 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 3617 . . . 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 3617 . . 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 7156 . 2 (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3836, 37impbid1 226 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 207  wa 396   = wceq 1530  wcel 2107  wral 3143  cop 4570  cfv 6352  (class class class)co 7148  Basecbs 16473  Hom chom 16566  compcco 16567  Catccat 16925  Monocmon 16988
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-cat 16929  df-mon 16990
This theorem is referenced by:  epii  17003  monsect  17043  fthmon  17187  setcmon  17337
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