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Theorem fthmon 17200
Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthmon.b 𝐵 = (Base‘𝐶)
fthmon.h 𝐻 = (Hom ‘𝐶)
fthmon.f (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
fthmon.x (𝜑𝑋𝐵)
fthmon.y (𝜑𝑌𝐵)
fthmon.r (𝜑𝑅 ∈ (𝑋𝐻𝑌))
fthmon.m 𝑀 = (Mono‘𝐶)
fthmon.n 𝑁 = (Mono‘𝐷)
fthmon.1 (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
Assertion
Ref Expression
fthmon (𝜑𝑅 ∈ (𝑋𝑀𝑌))

Proof of Theorem fthmon
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthmon.r . 2 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
2 eqid 2824 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2824 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2824 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
5 fthmon.n . . . . . 6 𝑁 = (Mono‘𝐷)
6 fthmon.f . . . . . . . . . . 11 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
7 fthfunc 17180 . . . . . . . . . . . 12 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 5114 . . . . . . . . . . 11 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
96, 8syl 17 . . . . . . . . . 10 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 5070 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 220 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 17136 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simprd 498 . . . . . . 7 (𝜑𝐷 ∈ Cat)
1514adantr 483 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16 fthmon.b . . . . . . . 8 𝐵 = (Base‘𝐶)
179adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
1816, 2, 17funcf1 17139 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19 fthmon.x . . . . . . . 8 (𝜑𝑋𝐵)
2019adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
2118, 20ffvelrnd 6855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑋) ∈ (Base‘𝐷))
22 fthmon.y . . . . . . . 8 (𝜑𝑌𝐵)
2322adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌𝐵)
2418, 23ffvelrnd 6855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑌) ∈ (Base‘𝐷))
25 simpr1 1190 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
2618, 25ffvelrnd 6855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑧) ∈ (Base‘𝐷))
27 fthmon.1 . . . . . . 7 (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
2827adantr 483 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
29 fthmon.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
3016, 29, 3, 17, 25, 20funcf2 17141 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
31 simpr2 1191 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
3230, 31ffvelrnd 6855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
33 simpr3 1192 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
3430, 33ffvelrnd 6855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑔) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 17009 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔)) ↔ ((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔)))
36 eqid 2824 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
371adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 17144 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)))
3916, 29, 36, 4, 17, 25, 20, 23, 33, 37funcco 17144 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔)))
4038, 39eqeq12d 2840 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) ↔ (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔))))
416adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
4213simpld 497 . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
4342adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 16959 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) ∈ (𝑧𝐻𝑌))
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 16959 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ∈ (𝑧𝐻𝑌))
4616, 29, 3, 41, 25, 23, 44, 45fthi 17191 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) ↔ (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
4740, 46bitr3d 283 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔)) ↔ (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
4816, 29, 3, 41, 25, 20, 31, 33fthi 17191 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
4935, 47, 483bitr3d 311 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ↔ 𝑓 = 𝑔))
5049biimpd 231 . . 3 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
5150ralrimivvva 3195 . 2 (𝜑 → ∀𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋)∀𝑔 ∈ (𝑧𝐻𝑋)((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
52 fthmon.m . . 3 𝑀 = (Mono‘𝐶)
5316, 29, 36, 52, 42, 19, 22ismon2 17007 . 2 (𝜑 → (𝑅 ∈ (𝑋𝑀𝑌) ↔ (𝑅 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋)∀𝑔 ∈ (𝑧𝐻𝑋)((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))))
541, 51, 53mpbir2and 711 1 (𝜑𝑅 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1536  wcel 2113  wral 3141  cop 4576   class class class wbr 5069  cfv 6358  (class class class)co 7159  Basecbs 16486  Hom chom 16579  compcco 16580  Catccat 16938  Monocmon 17001   Func cfunc 17127   Faith cfth 17176
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411  df-ixp 8465  df-cat 16942  df-mon 17003  df-func 17131  df-fth 17178
This theorem is referenced by:  fthepi  17201
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