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Theorem moni 16596
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 16594 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))))
101, 9mpbid 223 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = )))
1110simprd 485 . . 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 468 . . . . . 6 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15 simpr 473 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1615oveq1d 6885 . . . . . 6 ((𝜑𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
1714, 16eleqtrrd 2888 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18 moni.k . . . . . . . . 9 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1918adantr 468 . . . . . . . 8 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
2019, 16eleqtrrd 2888 . . . . . . 7 ((𝜑𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
2120adantr 468 . . . . . 6 (((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22 simpllr 784 . . . . . . . . . . 11 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑧 = 𝑍)
2322opeq1d 4601 . . . . . . . . . 10 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
2423oveq1d 6885 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (⟨𝑧, 𝑋· 𝑌) = (⟨𝑍, 𝑋· 𝑌))
25 eqidd 2807 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝐹 = 𝐹)
26 simplr 776 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → 𝑔 = 𝐺)
2724, 25, 26oveq123d 6891 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺))
28 simpr 473 . . . . . . . . 9 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → = 𝐾)
2924, 25, 28oveq123d 6891 . . . . . . . 8 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝐹(⟨𝑧, 𝑋· 𝑌)) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3027, 29eqeq12d 2821 . . . . . . 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 2821 . . . . . . 7 ((((𝜑𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ = 𝐾) → (𝑔 = 𝐺 = 𝐾))
3230, 31imbi12d 335 . . . . . 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 3505 . . . . 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 3503 . . . 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 3503 . . 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 6878 . 2 (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾))
3836, 37impbid1 216 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 197  wa 384   = wceq 1637  wcel 2156  wral 3096  cop 4376  cfv 6097  (class class class)co 6870  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16525  Monocmon 16588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-1st 7394  df-2nd 7395  df-cat 16529  df-mon 16590
This theorem is referenced by:  epii  16603  monsect  16643  fthmon  16787  setcmon  16937
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